数学研发网设为首页收藏本站
切换到窄版

 找回密码
 欢迎注册
数学研发论坛»论坛 › 【数学研究】 › 趣题妙解 › 训练营成员的能力值分布
123下一页
返回列表 发新帖
楼主: KeyTo9_Fans

[原创] 训练营成员的能力值分布

[复制链接]
KeyTo9_Fans
 楼主| 发表于 2021-10-25 15:00:53 | 显示全部楼层
精确计算了$3$个新人闯营的情况，结果如下：

=====

第$1$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1$，他的能力值的概率密度函数为$f(x)=1$，$0<x<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$2$。

=====

第$2$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1/4$，他的能力值的概率密度函数为$f(x)=2x$，$0<x<1$；

营中有$2$人的概率为$3/4$，他们的能力值的概率密度函数为$f(x,y)=2$，$0<x<y<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$11/4$。

=====

第$3$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1/12$，他的能力值的概率密度函数为$f(x)=3x^2$，$0<x<1$；

营中有$2$人的概率为$5/12$，他们的能力值的概率密度函数为$f(x,y)=(12x+9y)/5$，$0<x<y<1$；

营中有$3$人的概率为$1/2$，他们的能力值的概率密度函数为$f(x,y,z)=6$，$0<x<y<z<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$41/12$。

=====

各种多重积分、条件概率、边缘分布的处理太容易出错了，我花了$2$天的时间纠错，才得到上面的结果。

#####

依次类推，$4$人的情况初步计算结果如下：

=====

第$4$人闯营后，除去能力值为$0$的初始成员，

营中有$1$人的概率为$1/32$，他的能力值的概率密度函数为$f(x)=4x^3$，$0<x<1$；

营中有$2$人的概率为$31/144$，他们的能力值的概率密度函数为$f(x,y)=(84x^2+84xy+54y^2)/31$，$0<x<y<1$；

营中有$3$人的概率为$127/288$，他们的能力值的概率密度函数为$f(x,y,z)=(696x+528y+432z)/127$，$0<x<y<z<1$；

营中有$4$人的概率为$5/16$，他们的能力值的概率密度函数为$f(x,y,z,w)=24$，$0<x<y<z<w<1$；

加上能力值为$0$的初始成员，营中人数的期望值为$581/144$。

=====

这个结果有待验证。
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-26 08:36:05 | 显示全部楼层
也就是说Fans发现n个人闯营以后，还保留h个人的密度分布函数是h个变量的n-h次齐次函数，这个结论很有意思。
假设n个人闯营以后，留下h人的概率为$p_{n,h}$, 这h个人的密度分布函数是n-h次齐次函数$f_{n,h}(x_1,x_2,...,x_h), 0\lt x_1\lt x_2\lt\cdots\lt x_h$
那么第n+1个人闯营以后，有$1/{h+1}$的概率选择和$x_t$做比较 $0\le t\le h$.
   其中闯营失败的密度分布为$\int_0^{x_t} x_{h+1} f_{n,h}(x_1,x_2,...,x_h)d x_{h+1} =x_t f_{n,h}(x_1,x_2,...,x_h)$,是需要添加到n+1个人留h个人的密度分布函数中，正好是n-h+1次齐次函数。并且计算这个函数多所有变量的累次积分可以得到这部分总的概率。
   闯营成功，那么落在$(x_s, x_{s+1})$的密度分布为$f_{n,h}(x_1,x_2,...,x_s,x_{s+2},x_{s+3},...,x_{h+1})$, 需要将这个函数添加到n+1个人留h+1个人的密度函数中，这正好是一个(n+1)-(h+1)次齐次函数

所以逻辑代码如下

$f_{1,1}(x_1)=1$

for(n=2, any){
f_{n,1..n}=0
for(h=1,n-1){
   for(t=0,h){
      if(t>0) $f_{n,h}(x_1,x_2,...,x_h)+=\frac{x_t f_{n-1,h}(x_1,x_2,...,x_h)}{h+1}$
      for(s=t,h)$f_{n,h+1}(x_1,x_2,...,x_{h+1}) += \frac{f_{n-1,h}(x_1,x_2,...,x_s,x_{s+2},...,x_{h+1})}{h+1}$
   }
}
}
比如这样经过一轮后可以得出$f_{2,1}(x_1)=\frac{x_1}2, f_{2,2}(x_1,x_2)=\frac3 2$, 正好对应Fans的n=2时的密度函数和前面的概率的乘积

点评

KeyTo9_Fans
这个数学归纳好棒！剩下的问题就是要设计算法，把密度函数里的那[2的(n-1)次方]个系数给他算出来。不然手算太花时间了。  发表于 2021-10-26 10:53
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-26 11:55:57 | 显示全部楼层
Level 1:
        1

Level 2:
        +1/2*A
        3/2
       
1/4 3/4
Average 11/4
Level 3:
        +1/4*A^2
        +1*A+3/4*B
        3
       
1/12 5/12 1/2
Average 41/12
Level 4:
        +1/8*A^3
        +7/12*A^2+7/12*AB+3/8*B^2
        +29/12*A+11/6*B+3/2*C
        15/2
       
1/32 31/144 127/288 5/16
Average 581/144
Level 5:
        +1/16*A^4
        +23/72*A^3+7/18*A^2B+23/72*AB^2+3/16*B^3
        +227/144*A^2+79/48*AB+37/36*B^2+197/144*AC+37/36*BC+3/4*C^2
        +111/16*A+85/16*B+35/8*C+15/4*D
        45/2
       
1/80 481/4320 5321/17280 731/1920 3/16
Average 79823/17280
Level 6:
        +1/32*A^5
        +73/432*A^4+17/72*A^3B+17/72*A^2B^2+73/432*AB^3+3/32*B^4
        +1601/1728*A^3+43/36*A^2B+569/576*AB^2+119/216*B^3+215/216*A^2C+97/96*ABC+139/216*B^2C+1283/1728*AC^2+119/216*BC^2+3/8*C^3
        +1579/320*A^2+5117/960*AB+625/192*B^2+6413/1440*AC+1945/576*BC+115/48*C^2+11081/2880*AD+1685/576*BD+115/48*CD+15/8*D^2
        +927/40*A+1431/80*B+237/16*C+51/4*D+45/4*E
        315/4
       
1/192 1511/25920 4147/20736 86723/259200 5617/19200 7/64
Average 74579/14400
Level 7:
        +1/64*A^6
        +227/2592*A^5+175/1296*A^4B+17/108*A^3B^2+175/1296*A^2B^3+227/2592*AB^4+3/64*B^5
        +10643/20736*A^4+5297/6912*A^3B+1801/2304*A^2B^2+3827/6912*AB^3+373/1296*B^4+4409/6912*A^3C+2765/3456*A^2BC+4565/6912*AB^2C+163/432*B^3C+4091/6912*A^2C^2+1327/2304*ABC^2+163/432*B^2C^2+8129/20736*AC^3+373/1296*BC^3+3/16*C^4
        +58973/19200*A^3+29831/7200*A^2B+198479/57600*AB^2+4255/2304*B^3+149911/43200*A^2C+1457/400*ABC+245/108*B^2C+225181/86400*AC^2+13495/6912*BC^2+365/288*C^3+64813/21600*A^2D+10171/3200*ABD+425/216*B^2D+6347/2400*ACD+255/128*BCD+415/288*C^2D+358147/172800*AD^2+10775/6912*BD^2+365/288*CD^2+15/16*D^3
        +14143/800*A^2+15709/800*AB+18849/1600*B^2+39551/2400*AC+15113/1200*BC+559/64*C^2+13717/960*AD+6569/600*BD+577/64*CD+55/8*D^2+20261/1600*AE+7777/800*BE+513/64*CE+55/8*DE+45/8*E^2
        +177/2*A+2751/40*B+2289/40*C+791/16*D+175/4*E+315/8*F
        315
       
1/448 1123/36288 73321/580608 18865097/72576000 3745823/12096000 42019/201600 1/16
Average 46119277/8064000
Level 8:
        +1/128*A^7
        +697/15552*A^6+577/7776*A^5B+379/3888*A^4B^2+379/3888*A^3B^3+577/7776*A^2B^4+697/15552*AB^5+3/128*B^6
        +68249/248832*A^5+6289/13824*A^4B+3763/6912*A^3B^2+19445/41472*A^2B^3+24713/82944*AB^4+1151/7776*B^5+47005/124416*A^4C+3809/6912*A^3BC+287/512*A^2B^2C+1375/3456*AB^3C+1643/7776*B^4C+8551/20736*A^3C^2+2267/4608*A^2BC^2+1859/4608*AB^2C^2+625/2592*B^3C^2+41803/124416*A^2C^3+1085/3456*ABC^3+1643/7776*B^2C^3+50579/248832*AC^4+1151/7776*BC^4+3/32*C^5
        +2038051/1152000*A^4+9621679/3456000*A^3B+3324383/1152000*A^2B^2+7006873/3456000*AB^3+27685/27648*B^4+4025177/1728000*A^3C+146459/48000*A^2BC+40509/16000*AB^2C+38485/27648*B^3C+3798137/1728000*A^2C^2+35459/16000*ABC^2+39175/27648*B^2C^2+7483547/5184000*AC^3+88765/82944*BC^3+1135/1728*C^4+6952669/3456000*A^3D+4604239/1728000*A^2BD+7650559/3456000*AB^2D+33325/27648*B^3D+1920757/864000*A^2CD+54997/24000*ABCD+19985/13824*B^2CD+2883317/1728000*AC^2D+34345/27648*BC^2D+475/576*C^3D+6063479/3456000*A^2D^2+2066633/1152000*ABD^2+31415/27648*B^2D^2+2565887/1728000*ACD^2+30625/27648*BCD^2+475/576*C^2D^2+11322389/10368000*AD^3+67445/82944*BD^3+1135/1728*CD^3+15/32*D^4
        +184747/16000*A^3+96977/6000*A^2B+648293/48000*AB^2+224321/32000*B^3+108773/8000*A^2C+88049/6000*ABC+143591/16000*B^2C+1479157/144000*AC^2+1117187/144000*BC^2+3721/768*C^3+113267/9600*A^2D+308227/24000*ABD+374923/48000*B^2D+257573/24000*ACD+195529/24000*BCD+553/96*C^2D+472673/57600*AD^2+895987/144000*BD^2+3907/768*CD^2+173/48*D^3+167329/16000*A^2E+45813/4000*ABE+333043/48000*B^2E+689767/72000*ACE+131123/18000*BCE+41/8*C^2E+59633/7200*ADE+909359/144000*BDE+663/128*CDE+193/48*D^2E+216919/32000*AE^2+1933/375*BE^2+1081/256*CE^2+173/48*DE^2+45/16*E^3
        +120241/1680*A^2+455709/5600*AB+7217/150*B^2+767897/11200*AC+84413/1600*BC+28721/800*C^2+801481/13440*AD+14721/320*BD+365407/9600*CD+5453/192*D^2+1186949/22400*AE+393127/9600*BE+162841/4800*CE+1869/64*DE+70/3*E^2+1606589/33600*AF+355229/9600*BF+98189/3200*CF+5075/192*DF+70/3*EF+315/16*F^2
        +21321/56*A+4167/14*B+2487/10*C+4311/20*D+765/4*E+345/2*F+315/2*G
        2835/2
       
1/1024 1133/68040 1569971/19906560 368368771/1935360000 540359959/1935360000 233002181/903168000 176651/1254400 9/256
Average 95151902939/15240960000
Level 9:
        +1/256*A^8
        +2123/93312*A^7+617/15552*A^6B+445/7776*A^5B^2+379/5832*A^4B^3+445/7776*A^3B^4+617/15552*A^2B^5+2123/93312*AB^6+3/256*B^7
        +427787/2985984*A^6+255307/995328*A^5B+172847/497664*A^4B^2+174581/497664*A^3B^3+264665/995328*A^2B^4+155579/995328*AB^5+3517/46656*B^6+634489/2985984*A^5C+21521/62208*A^4BC+7631/18432*A^3B^2C+7399/20736*A^2B^3C+225715/995328*AB^4C+5345/46656*B^5C+391957/1492992*A^4C^2+60359/165888*A^3BC^2+6709/18432*A^2B^2C^2+43231/165888*AB^3C^2+6793/46656*B^4C^2+376351/1492992*A^3C^3+17759/62208*A^2BC^3+115541/497664*AB^2C^3+6793/46656*B^3C^3+550267/2985984*A^2C^4+55177/331776*ABC^4+5345/46656*B^2C^4+310937/2985984*AC^5+3517/46656*BC^5+3/64*C^6
        +67112237/69120000*A^5+19662013/11520000*A^4B+36057953/17280000*A^3B^2+186952007/103680000*A^2B^3+233729351/207360000*AB^4+175135/331776*B^5+110806219/77760000*A^4C+37739581/17280000*A^3BC+12984797/5760000*A^2B^2C+13721221/8640000*AB^3C+399965/497664*B^4C+41214317/25920000*A^3C^2+8575429/4320000*A^2BC^2+28277213/17280000*AB^2C^2+77885/82944*B^3C^2+201980873/155520000*A^2C^3+7257239/5760000*ABC^3+411035/497664*B^2C^3+240223189/311040000*AC^4+565255/995328*BC^4+3485/10368*C^5+382093421/311040000*A^4D+32924701/17280000*A^3BD+7577083/3840000*A^2B^2D+23979307/17280000*AB^3D+345505/497664*B^4D+27447301/17280000*A^3CD+654743/320000*A^2BCD+3257369/1920000*AB^2CD+26165/27648*B^3CD+8629687/5760000*A^2C^2D+949573/640000*ABC^2D+5905/6144*B^2C^2D+6391187/6480000*AC^3D+30145/41472*BC^3D+4715/10368*C^4D+65770319/51840000*A^3D^2+55695587/34560000*A^2BD^2+46011541/34560000*AB^2D^2+62395/82944*B^3D^2+11564453/8640000*A^2CD^2+1282187/960000*ABCD^2+47605/55296*B^2CD^2+17234033/17280000*AC^2D^2+40685/55296*BC^2D^2+1765/3456*C^3D^2+307711331/311040000*A^2D^3+16852987/17280000*ABD^3+314695/497664*B^2D^3+41665211/51840000*ACD^3+24635/41472*BCD^3+4715/10368*C^2D^3+352248043/622080000*AD^4+416255/995328*BD^4+3485/10368*CD^4+15/64*D^5
        +19808767/2880000*A^4+451999/40000*A^3B+2848949/240000*A^2B^2+23861491/2880000*AB^3+7536727/1920000*B^4+41027321/4320000*A^3C+2321627/180000*A^2BC+429993/40000*AB^2C+49476287/8640000*B^3C+19631773/2160000*A^2C^2+2265593/240000*ABC^2+12779393/2160000*B^2C^2+51225959/8640000*AC^3+38295619/8640000*BC^3+23875/9216*C^4+5689987/691200*A^3D+97596917/8640000*A^2BD+162882827/17280000*AB^2D+671833/135000*B^3D+20444047/2160000*A^2CD+481169/48000*ABCD+53665769/8640000*B^2CD+61707479/8640000*AC^2D+92612789/17280000*BC^2D+10529/3072*C^3D+25182503/3456000*A^2D^2+14724173/1920000*ABD^2+41168197/8640000*B^2D^2+6120601/960000*ACD^2+27611413/5760000*BCD^2+10699/3072*C^2D^2+15565891/3456000*AD^3+29217719/8640000*BD^3+25321/9216*CD^3+535/288*D^4+13994189/1920000*A^3E+87056443/8640000*A^2BE+145419433/17280000*AB^2E+4237973/960000*B^3E+1521013/180000*A^2CE+6479063/720000*ABCE+1777817/320000*B^2CE+18383579/2880000*AC^2E+9211831/1920000*BC^2E+9353/3072*C^3E+6318383/864000*A^2DE+11314307/1440000*ABDE+41668459/8640000*B^2DE+9425773/1440000*ACDE+1780571/360000*BCDE+1815/512*C^2DE+4395001/864000*AD^2E+66332929/17280000*BD^2E+9605/3072*CD^2E+217/96*D^3E+11579249/1920000*A^2E^2+12305867/1920000*ABE^2+949717/240000*B^2E^2+46100767/8640000*ACE^2+69498769/17280000*BCE^2+2969/1024*C^2E^2+3970739/864000*ADE^2+60028769/17280000*BDE^2+8701/3072*CDE^2+217/96*D^2E^2+20529959/5760000*AE^3+3865919/1440000*BE^3+6715/3072*CE^3+535/288*DE^3+45/32*E^4
        +3819141/78400*A^3+41358923/588000*A^2B+138807691/2352000*AB^2+238623/8000*B^3+419114779/7056000*A^2C+771971/11760*ABC+11387551/288000*B^2C+70724951/1568000*AC^2+137249/4000*BC^2+995827/48000*C^3+29186489/564480*A^2D+40641701/705600*ABD+994133/28800*B^2D+68187319/1411200*ACD+531769/14400*BCD+7376929/288000*C^2D+68078057/1881600*AD^2+318269/11520*BD^2+13075559/576000*CD^2+35735/2304*D^3+216150841/4704000*A^2E+363254653/7056000*ABE+2213351/72000*B^2E+20333851/470400*ACE+264719/8000*BCE+6579629/288000*C^2E+52901251/1411200*ADE+2071517/72000*BDE+1138039/48000*CDE+1729/96*D^2E+282156011/9408000*AE^2+13227809/576000*BE^2+2722321/144000*CE^2+12439/768*DE^2+875/72*E^3+97507937/2352000*A^2F+36572511/784000*ABF+888937/32000*B^2F+1920637/49000*ACF+120169/4000*BCF+1984213/96000*C^2F+959935/28224*ADF+7527037/288000*BDF+6208069/288000*CDF+4697/288*D^2F+14186059/470400*AEF+742959/32000*BEF+5522069/288000*CEF+2107/128*DEF+1925/144*E^2F+119764997/4704000*AF^2+1250417/64000*BF^2+3092593/192000*CF^2+31829/2304*DF^2+875/72*EF^2+315/32*F^3
        +5074911/15680*A^2+1177341/3136*AB+107439/490*B^2+4977001/15680*AC+963259/3920*BC+8227/50*C^2+4341053/15680*AD+1684409/7840*BD+28563/160*CD+52299/400*D^2+19331701/78400*AE+7514061/39200*BE+127571/800*CE+13771/100*DE+1725/16*E^2+7491011/33600*AF+10204267/58800*BF+57801/400*CF+37463/300*DF+5305/48*EF+365/4*F^2+9591161/47040*AG+2335451/14700*BG+26477/200*CG+68681/600*DG+4865/48*EG+365/4*FG+315/4*G^2
        +116685/64*A+641925/448*B+134595/112*C+16713/16*D+29727/32*E+26865/32*F+12285/16*G+2835/4*H
        14175/2
       
1/2304 2133889/235146240 738921023/15049359360 20159805067/149299200000 362925784139/1567641600000 1353647390737/5120962560000 17026982389/85349376000 1652579/18063360 5/256
Average 3112703236301951/460886630400000
Level 10:
        +1/512*A^9
        +6433/559872*A^8+5825/279936*A^7B+1507/46656*A^6B^2+2851/69984*A^5B^3+2851/69984*A^4B^4+1507/46656*A^3B^5+5825/279936*A^2B^6+6433/559872*AB^7+3/512*B^8
        +2642081/35831808*A^7+416891/2985984*A^6B+828841/3981312*A^5B^2+357595/1492992*A^4B^3+841667/3981312*A^3B^4+16061/110592*A^2B^5+963569/11943936*AB^6+10679/279936*B^7+344545/2985984*A^6C+1216709/5971968*A^5BC+15307/55296*A^4B^2C+279097/995328*A^3B^3C+7829/36864*A^2B^4C+742981/5971968*AB^5C+5665/93312*B^6C+1874083/11943936*A^5C^2+362923/1492992*A^4BC^2+189419/663552*A^3B^2C^2+13567/55296*A^2B^3C^2+1890055/11943936*AB^4C^2+7849/93312*B^5C^2+770279/4478976*A^4C^3+672899/2985984*A^3BC^3+36563/165888*A^2B^2C^3+476539/2985984*AB^3C^3+26443/279936*B^4C^3+1758649/11943936*A^3C^4+474823/2985984*A^2BC^4+1531919/11943936*AB^2C^4+7849/93312*B^3C^4+294277/2985984*A^2C^5+172103/1990656*ABC^5+5665/93312*B^2C^5+1896203/35831808*AC^6+10679/279936*BC^6+3/128*C^7
        +2142181219/4147200000*A^6+2449575193/2488320000*A^5B+1700139629/1244160000*A^4B^2+346342243/248832000*A^3B^3+2616229451/2488320000*A^2B^4+7521555337/12441600000*AB^5+1088125/3981312*B^6+3060363737/3732480000*A^5C+48698593/34560000*A^4BC+3960593/2304000*A^3B^2C+51353867/34560000*A^2B^3C+10720117/11520000*AB^4C+5315045/11943936*B^5C+1939390391/1866240000*A^4C^2+10476359/6912000*A^3BC^2+17730901/11520000*A^2B^2C^2+37784737/34560000*AB^3C^2+3469505/5971968*B^4C^2+375203677/373248000*A^3C^3+10318103/8640000*A^2BC^3+4219009/4320000*AB^2C^3+3501275/5971968*B^3C^3+2730124469/3732480000*A^2C^4+11859983/17280000*ABC^4+5491895/11943936*B^2C^4+7539187643/18662400000*AC^5+3526645/11943936*BC^5+10615/62208*C^6+1053290173/1492992000*A^5D+95456557/77760000*A^4BD+69283873/46080000*A^3B^2D+33707017/25920000*A^2B^3D+2020275109/2488320000*AB^4D+4580125/11943936*B^5D+317676443/311040000*A^4CD+2965949/1920000*A^3BCD+18363161/11520000*A^2B^2CD+19418239/17280000*AB^3CD+143165/248832*B^4CD+236874551/207360000*A^3C^2D+12118537/8640000*A^2BC^2D+39935927/34560000*AB^2C^2D+446435/663552*B^3C^2D+9088777/9720000*A^2C^3D+7701377/8640000*ABC^3D+147305/248832*B^2C^3D+76870223/138240000*AC^4D+539605/1327104*BC^4D+15185/62208*C^5D+3088982429/3732480000*A^4D^2+510113609/414720000*A^3BD^2+771817/614400*A^2B^2D^2+368981497/414720000*AB^3D^2+2771785/5971968*B^4D^2+70574509/69120000*A^3CD^2+350119/276480*A^2BCD^2+4513901/4320000*AB^2CD^2+133345/221184*B^3CD^2+97157/102400*A^2C^2D^2+10488791/11520000*ABC^2D^2+44425/73728*B^2C^2D^2+43488217/69120000*AC^3D^2+101705/221184*BC^3D^2+18805/62208*C^4D^2+2861696663/3732480000*A^3D^3+144530663/155520000*A^2BD^3+947952983/1244160000*AB^2D^3+2682235/5971968*B^3D^3+26578009/34560000*A^2CD^3+1068089/1440000*ABCD^3+4505/9216*B^2CD^3+352874669/622080000*AC^2D^3+826025/1990656*BC^2D^3+18805/62208*C^3D^3+4041994067/7464960000*A^2D^4+85710839/165888000*ABD^4+4086055/11943936*B^2D^4+105597821/248832000*ACD^4+1239175/3981312*BCD^4+15185/62208*C^2D^4+10834967141/37324800000*AD^5+2545085/11943936*BD^5+10615/62208*CD^5+15/128*D^6
        +222624443/57600000*A^5+615679709/86400000*A^4B+1773963/200000*A^3B^2+221382803/28800000*A^2B^3+819124517/172800000*AB^4+243683849/115200000*B^5+774600073/129600000*A^4C+2474591297/259200000*A^3BC+862054099/86400000*A^2B^2C+905321087/129600000*AB^3C+1760808077/518400000*B^4C+880557047/129600000*A^3C^2+570944753/64800000*A^2BC^2+1891271791/259200000*AB^2C^2+1049246279/259200000*B^3C^2+1441659881/259200000*A^2C^3+160361221/28800000*ABC^3+925887271/259200000*B^2C^3+1690486633/518400000*AC^4+1253575553/518400000*BC^4+149689/110592*C^5+67026047/12960000*A^4D+4330588873/518400000*A^3BD+378340849/43200000*A^2B^2D+3173575151/518400000*AB^3D+1526746627/518400000*B^4D+3625850677/518400000*A^3CD+161163329/17280000*A^2BCD+268363369/34560000*AB^2CD+544606229/129600000*B^3CD+216514507/32400000*A^2C^2D+78466901/11520000*ABC^2D+560714519/129600000*B^2C^2D+2266066769/518400000*AC^3D+420903853/129600000*BC^3D+107273/55296*C^4D+17642503/3240000*A^3D^2+3720984223/518400000*A^2BD^2+3087846839/518400000*AB^2D^2+844274929/259200000*B^3D^2+344851813/57600000*A^2CD^2+4248131/691200*ABCD^2+335769671/86400000*B^2CD^2+2325185801/518400000*AC^2D^2+432307417/129600000*BC^2D^2+20513/9216*C^3D^2+440966329/103680000*A^2D^3+373565461/86400000*ABD^3+711608621/259200000*B^2D^3+309216409/86400000*ACD^3+230383427/86400000*BCD^3+109967/55296*C^2D^3+497423717/207360000*AD^4+926193253/518400000*BD^4+159739/110592*CD^4+1637/1728*D^5+14631369/3200000*A^4E+3859458217/518400000*A^3BE+225297329/28800000*A^2B^2E+2831490359/518400000*AB^3E+450598739/172800000*B^4E+3234202801/518400000*A^3CE+2170533059/259200000*A^2BCE+3616971899/518400000*AB^2CE+486553559/129600000*B^3CE+774341119/129600000*A^2C^2E+1058154703/172800000*ABC^2E+1004443723/259200000*B^2C^2E+2023607737/518400000*AC^3E+47058361/16200000*BC^3E+10565/6144*C^4E+279624421/51840000*A^3DE+948410129/129600000*A^2BDE+395593511/64800000*AB^2DE+210889417/64800000*B^3DE+132062761/21600000*A^2CDE+460463/72000*ABCDE+345471107/86400000*B^2CDE+1196102821/259200000*AC^2DE+446778139/129600000*BC^2DE+2291/1024*C^3DE+495514561/103680000*A^2D^2E+23834291/4800000*ABD^2E+403576549/129600000*B^2D^2E+2133702847/518400000*ACD^2E+1597144331/518400000*BCD^2E+4647/2048*C^2D^2E+102349489/34560000*AD^3E+127498703/57600000*BD^3E+8257/4608*CD^3E+2129/1728*D^4E+7205773/1600000*A^3E^2+3115001977/518400000*A^2BE^2+2589001661/518400000*AB^2E^2+77822921/28800000*B^3E^2+2601427381/518400000*A^2CE^2+448942979/86400000*ABCE^2+52922713/16200000*B^2CE^2+1952710453/518400000*AC^2E^2+727853869/259200000*BC^2E^2+1895/1024*C^3E^2+448769117/103680000*A^2DE^2+24429883/5400000*ABDE^2+366319529/129600000*B^2DE^2+648431989/172800000*ACDE^2+486055517/172800000*BCDE^2+12673/6144*C^2DE^2+51629833/17280000*AD^2E^2+386455271/172800000*BD^2E^2+11129/6144*CD^2E^2+787/576*D^3E^2+64894429/19200000*A^2E^3+300248999/86400000*ABE^3+47306413/21600000*B^2E^3+746686981/259200000*ACE^3+557888141/259200000*BCE^3+29321/18432*C^2E^3+256435079/103680000*ADE^3+960691907/518400000*BDE^3+6923/4608*CDE^3+2129/1728*D^2E^3+70949737/38400000*AE^4+119280403/86400000*BE^4+41233/36864*CE^4+1637/1728*DE^4+45/64*E^5
        +8856743221/296352000*A^4+1047672859/20580000*A^3B+371065453/6860000*A^2B^2+4121249491/109760000*AB^3+74233159/4320000*B^4+127294670959/2963520000*A^3C+2971240577/49392000*A^2BC+4971978127/98784000*AB^2C+451165183/17280000*B^3C+15413234873/370440000*A^2C^2+364866619/8232000*ABC^2+118021267/4320000*B^2C^2+53326454777/1975680000*AC^3+58542631/2880000*BC^3+32840549/2880000*C^4+88554278909/2370816000*A^3D+187844568017/3556224000*A^2BD+314702151107/7112448000*AB^2D+78665657/3456000*B^3D+9874322329/222264000*A^2CD+4757656667/98784000*ABCD+304739939/10368000*B^2CD+119782613927/3556224000*AC^2D+527811683/20736000*BC^2D+545002759/34560000*C^3D+79393666637/2370816000*A^2D^2+28547325041/790272000*ABD^2+15266209/691200*B^2D^2+2382640441/79027200*ACD^2+157779559/6912000*BCD^2+561439879/34560000*C^2D^2+48797466799/2370816000*AD^3+3586541/230400*BD^3+439324333/34560000*CD^3+227045/27648*D^4+131027595607/3951360000*A^3E+839656033991/17781120000*A^2BE+1407896870021/35562240000*AB^2E+233224943/11520000*B^3E+70680573317/1778112000*A^2CE+3565788053/82320000*ABCE+1365802207/51840000*B^2CE+107288246279/3556224000*AC^2E+2367475537/103680000*BC^2E+242702467/17280000*C^3E+613561391/17781120*A^2DE+9364143823/246960000*ABDE+1189147729/51840000*B^2DE+15660873289/493920000*ACDE+34727819/1440000*BCDE+293077379/17280000*C^2DE+85830883163/3556224000*AD^2E+1901354539/103680000*BD^2E+518720699/34560000*CD^2E+96551/9216*D^3E+109885548667/3951360000*A^2E^2+119631199379/3951360000*ABE^2+23555637/1280000*B^2E^2+29994182833/1185408000*ACE^2+221234363/11520000*BCE^2+117228251/8640000*C^2E^2+5183142031/237081600*ADE^2+575008483/34560000*BDE^2+52341961/3840000*CDE^2+10885/1024*D^2E^2+64568415599/3951360000*AE^3+428455783/34560000*BE^3+43827301/4320000*CE^3+79765/9216*DE^3+2695/432*E^4+59049907829/1975680000*A^3F+63399864529/1481760000*A^2BF+106374042349/2963520000*AB^2F+210510461/11520000*B^3F+320427028333/8890560000*A^2CF+2163978473/54880000*ABCF+620049227/25920000*B^2CF+243385854659/8890560000*AC^2F+268861943/12960000*BC^2F+48732061/3840000*C^3F+55661198107/1778112000*A^2DF+102335894089/2963520000*ABDF+67516631/3240000*B^2DF+85629860941/2963520000*ACDF+95047603/4320000*BCDF+44418899/2880000*C^2DF+77936969611/3556224000*AD^2F+432138329/25920000*BD^2F+78662269/5760000*CD^2F+87311/9216*D^3F+5485369199/197568000*A^2EF+91306886897/2963520000*ABEF+35562569/1920000*B^2EF+19118359817/740880000*ACEF+170049523/8640000*BCEF+237257419/17280000*C^2EF+13234191319/592704000*ADEF+29507513/1728000*BDEF+5050843/360000*CDEF+49763/4608*D^2EF+99511581/5488000*AE^2F+53120669/3840000*BE^2F+392371189/34560000*CE^2F+89411/9216*DE^2F+2135/288*E^3F+46688629109/1975680000*A^2F^2+4265984753/164640000*ABF^2+180622421/11520000*B^2F^2+892257419/41160000*ACF^2+15825593/960000*BCF^2+44472701/3840000*C^2F^2+22221314009/1185408000*ADF^2+246990611/17280000*BDF^2+25326469/2160000*CDF^2+83741/9216*D^2F^2+1091572333/65856000*AEF^2+16208423/1280000*BEF^2+359474129/34560000*CEF^2+9107/1024*DEF^2+2135/288*E^2F^2+79185031019/5927040000*AF^3+351154811/34560000*BF^3+95943491/11520000*CF^3+196679/27648*DF^3+2695/432*EF^3+315/64*F^4
        +1002556341/4390400*A^3+13936511/41160*A^2B+751050577/2634240*AB^2+19330809/137200*B^3+354276991/1234800*A^2C+426259583/1317120*ABC+950506139/4939200*B^2C+2882714773/13171200*AC^2+55171901/329280*BC^2+591323/6000*C^3+2474013949/9878400*A^2D+11157703/39200*ABD+1664085959/9878400*B^2D+788958829/3292800*ACD+606610997/3292800*BCD+603329/4800*C^2D+773609327/4390400*AD^2+148604597/1097600*BD^2+357557/3200*CD^2+592071/8000*D^3+114781657/514500*A^2E+2098492619/8232000*ABE+2475591463/16464000*B^2E+707218409/3292800*ACE+2723026343/16464000*BCE+2696617/24000*C^2E+410006529/2195200*ADE+395661551/2744000*BDE+2863663/24000*CDE+355841/4000*D^2E+9648881813/65856000*AE^2+1858499539/16464000*BE^2+4479887/48000*CE^2+962729/12000*DE^2+3725/64*E^3+948719411/4704000*A^2F+3809645143/16464000*ABF+560365061/4116000*B^2F+9635907137/49392000*ACF+3714590141/24696000*BCF+407389/4000*C^2F+419210389/2469600*ADF+6480293077/49392000*BDF+1564697/14400*CDF+2905519/36000*D^2F+62094107/411600*AEF+1923197467/16464000*BEF+1743539/18000*CEF+166843/2000*DEF+1195/18*E^2F+391112467/3136000*AF^2+264218877/2744000*BF^2+637799/8000*CF^2+2469737/36000*DF^2+34855/576*EF^2+1135/24*F^3+404780547/2195200*A^2G+1164707353/5488000*ABG+21372693/171500*B^2G+2947586711/16464000*ACG+1137365279/8232000*BCG+186623/2000*C^2G+96219211/617400*ADG+1191026309/9878400*BDG+7194251/72000*CDG+2663879/36000*D^2G+38019839/274400*AEG+117864889/1097600*BEG+3207731/36000*CEG+57583/750*DEG+4385/72*E^2G+117671677/940800*AFG+799062559/8232000*BFG+644957/8000*CFG+834203/12000*DFG+1965/32*EFG+1235/24*F^2G+1423464611/13171200*AG^2+86003999/1029000*BG^2+277123/4000*CG^2+1074011/18000*DG^2+30335/576*EG^2+1135/24*FG^2+315/8*G^3
        +5800951/3584*A^2+47928733/25088*AB+27700299/25088*B^2+725873/448*AC+31639677/25088*BC+40806/49*C^2+177722663/125440*AD+138655257/125440*BD+3613941/3920*CD+212607/320*D^2+396554453/313600*AE+619798257/627200*BE+129385917/156800*CE+2286909/3200*DE+351783/640*E^2+102627457/89600*AF+2294349/2560*BF+29361933/39200*CF+2077293/3200*DF+368553/640*EF+59715/128*F^2+32901383/31360*AG+515390343/627200*BG+10775997/15680*CG+953481/1600*DG+169239/320*EG+61065/128*FG+12915/32*G^2+121610249/125440*AH+95325147/125440*BH+6231951/9800*CH+110331/200*DH+313443/640*EH+56565/128*FH+12915/32*GH+2835/8*H^2
        +153955/16*A+486275/64*B+2864475/448*C+624075/112*D+79455/16*E+143865/32*F+131775/32*G+15225/4*H+14175/4*I
        155925/4
       
1/5120 35303591/7054387200 493315123/16124313600 177040148897/1881169920000 4913011142447/26873856000000 15756307796485993/64524128256000000 20497966660523/89616844800000 166085069503/1137991680000 3749981/65028096 11/1024
Average 1403936289503846939/193572384768000000
Level 11:
        +1/1024*A^10
        +19427/3359232*A^9+18083/1679616*A^8B+14867/839808*A^7B^2+10223/419904*A^6B^3+2851/104976*A^5B^4+10223/419904*A^4B^5+14867/839808*A^3B^6+18083/1679616*A^2B^7+19427/3359232*AB^8+3/1024*B^9
        +16160483/429981696*A^8+393599/5308416*A^7B+1899085/15925248*A^6B^2+21880697/143327232*A^5B^3+21996131/143327232*A^4B^4+5802757/47775744*A^3B^5+11076871/143327232*A^2B^6+5904467/143327232*AB^7+32293/1679616*B^8+26294663/429981696*A^7C+2739581/23887872*A^6BC+8226253/47775744*A^5B^2C+2379059/11943936*A^4B^3C+934529/5308416*A^3B^4C+2878573/23887872*A^2B^5C+352739/5308416*AB^6C+53161/1679616*B^7C+12843125/143327232*A^6C^2+7210885/47775744*A^5BC^2+4809619/23887872*A^4B^2C^2+1614863/7962624*A^3B^3C^2+7357123/47775744*A^2B^4C^2+4380689/47775744*AB^5C^2+26299/559872*B^6C^2+47031139/429981696*A^5C^3+5736793/35831808*A^4BC^3+4366837/23887872*A^3B^2C^3+1867291/11943936*A^2B^3C^3+14773337/143327232*AB^4C^3+97793/1679616*B^5C^3+45992233/429981696*A^4C^4+6349537/47775744*A^3BC^4+2021249/15925248*A^2B^2C^4+4442747/47775744*AB^3C^4+97793/1679616*B^4C^4+11893607/143327232*A^3C^5+2054509/23887872*A^2BC^5+3289657/47775744*AB^2C^5+26299/559872*B^3C^5+22247381/429981696*A^2C^6+2120323/47775744*ABC^6+53161/1679616*B^2C^6+11501729/429981696*AC^7+32293/1679616*BC^7+3/256*C^8
        +66988690253/248832000000*A^7+101620711991/186624000000*A^6B+41530314679/49766400000*A^5B^2+18117943157/18662400000*A^4B^3+42730073149/49766400000*A^3B^4+36412749467/62208000000*A^2B^5+236453710919/746496000000*AB^6+6681655/47775744*B^7+84445174591/186624000000*A^6C+10530926797/12441600000*A^5BC+7314046697/6220800000*A^4B^2C+2487842299/2073600000*A^3B^3C+11271540647/12441600000*A^2B^4C+16187586161/31104000000*AB^5C+2857895/11943936*B^6C+47239981201/74649600000*A^5C^2+3222000031/3110400000*A^4BC^2+2571735091/2073600000*A^3B^2C^2+92364091/86400000*A^2B^3C^2+8450480711/12441600000*AB^4C^2+16308455/47775744*B^5C^2+19612923203/27993600000*A^4C^3+1506025823/1555200000*A^3BC^3+62363867/64800000*A^2B^2C^3+4297546999/6220800000*AB^3C^3+6947605/17915904*B^4C^3+44810829331/74649600000*A^3C^4+4238981681/6220800000*A^2BC^4+3438940243/6220800000*AB^2C^4+16539245/47775744*B^3C^4+37161700613/93312000000*A^2C^5+7539040049/20736000000*ABC^5+2962475/11943936*B^2C^5+233107892341/1119744000000*AC^6+21713695/143327232*BC^6+32165/373248*C^7+2686276423/6912000000*A^6D+54964658707/74649600000*A^5BD+236653231/230400000*A^4B^2D+13057159579/12441600000*A^3B^3D+1094899313/1382400000*A^2B^4D+169190600917/373248000000*AB^5D+819095/3981312*B^6D+5703407149/9331200000*A^5CD+89904097/86400000*A^4BCD+293131153/230400000*A^3B^2CD+17833361/16200000*A^2B^3CD+2854565599/4147200000*AB^4CD+7923545/23887872*B^5CD+7268848133/9331200000*A^4C^2D+1162332701/1036800000*A^3BC^2D+65564087/57600000*A^2B^2C^2D+1677707579/2073600000*AB^3C^2D+2597965/5971968*B^4C^2D+14122403431/18662400000*A^3C^3D+343952369/388800000*A^2BC^3D+1124572813/1555200000*AB^2C^3D+5259095/11943936*B^3C^3D+10277426483/18662400000*A^2C^4D+2107085839/4147200000*ABC^4D+4117945/11943936*B^2C^4D+56553413621/186624000000*AC^5D+5262425/23887872*BC^5D+15985/124416*C^6D+75060209767/149299200000*A^5D^2+15664322771/18662400000*A^4BD^2+8394382889/8294400000*A^3B^2D^2+1810842119/2073600000*A^2B^3D^2+82388159717/149299200000*AB^4D^2+12987055/47775744*B^5D^2+4326797327/6220800000*A^4CD^2+2099384147/2073600000*A^3BCD^2+26426707/25600000*A^2B^2CD^2+379593901/518400000*AB^3CD^2+774845/1990656*B^4CD^2+1057766129/1382400000*A^3C^2D^2+34776863/38400000*A^2BC^2D^2+171116969/230400000*AB^2C^2D^2+395485/884736*B^3C^2D^2+1944876331/3110400000*A^2C^3D^2+44403287/76800000*ABC^3D^2+780755/1990656*B^2C^3D^2+28115060831/74649600000*AC^4D^2+13057315/47775744*BC^4D^2+21445/124416*C^5D^2+29887646651/55987200000*A^4D^3+28151338237/37324800000*A^3BD^3+1565154029/2073600000*A^2B^2D^3+20149731719/37324800000*AB^3D^3+5311955/17915904*B^4D^3+2246519/3600000*A^3CD^3+576427253/777600000*A^2BCD^3+3778960367/6220800000*AB^2CD^3+161245/442368*B^3CD^3+1764285731/3110400000*A^2C^2D^3+1092544429/2073600000*ABC^2D^3+709405/1990656*B^2C^2D^3+7083096107/18662400000*AC^3D^3+3288295/11943936*BC^3D^3+71575/373248*C^4D^3+66540440197/149299200000*A^3D^4+19287919271/37324800000*A^2BD^4+62755700737/149299200000*AB^2D^4+12333085/47775744*B^3D^4+1768713599/4147200000*A^2CD^4+275654609/691200000*ABCD^4+355915/1327104*B^2CD^4+23236508191/74649600000*AC^2D^4+10803275/47775744*BC^2D^4+21445/124416*C^3D^4+54035328101/186624000000*A^2D^5+33449525819/124416000000*ABD^5+2164345/11943936*B^2D^5+1284717763/5832000000*ACD^5+3836585/23887872*BCD^5+15985/124416*C^2D^5+330648465067/2239488000000*AD^6+15464375/143327232*BD^6+32165/373248*CD^6+15/256*D^7
        +21674001823/10368000000*A^6+217440397/51840000*A^5B+1541085677/259200000*A^4B^2+79055291/12960000*A^3B^3+2381825393/518400000*A^2B^4+26889613579/10368000000*AB^5+7684753063/6912000000*B^6+3639117827/1036800000*A^5C+9815416603/1555200000*A^4BC+12171930137/1555200000*A^3B^2C+421956517/62208000*A^2B^3C+6524721419/1555200000*AB^4C+3974137693/2073600000*B^5C+7069469641/1555200000*A^4C^2+5381204899/777600000*A^3BC^2+409993409/57600000*A^2B^2C^2+7811723941/1555200000*AB^3C^2+7962031387/3110400000*B^4C^2+172066429/38880000*A^3C^3+4255877627/777600000*A^2BC^3+6994327883/1555200000*AB^2C^3+404928401/155520000*B^3C^3+23095157/7200000*A^2C^4+50381243/16200000*ABC^4+2111244413/1036800000*B^2C^4+54060656471/31104000000*AC^5+39844353511/31104000000*BC^5+924619/1327104*C^6+12572426011/4147200000*A^5D+17153695309/3110400000*A^4BD+42707065829/6220800000*A^3B^2D+18516506651/3110400000*A^2B^3D+45678792677/12441600000*AB^4D+190985231/115200000*B^5D+4778402209/1036800000*A^4CD+3757559899/518400000*A^3BCD+7843606219/1036800000*A^2B^2CD+549800599/103680000*AB^3CD+1355352799/518400000*B^4CD+8152074731/1555200000*A^3C^2D+20760373871/3110400000*A^2BC^2D+2147012021/388800000*AB^2C^2D+3872719889/1244160000*B^3C^2D+891384433/207360000*A^2C^3D+547487101/129600000*ABC^3D+355997089/129600000*B^2C^3D+581451143/230400000*AC^4D+7715281169/4147200000*BC^4D+1406753/1327104*C^5D+4524124681/1244160000*A^4D^2+11685895049/2073600000*A^3BD^2+4028589061/691200000*A^2B^2D^2+8505139249/2073600000*AB^3D^2+3194809141/1555200000*B^4D^2+14613914999/3110400000*A^3CD^2+1878448741/311040000*A^2BCD^2+1556402237/311040000*AB^2CD^2+3479851669/1244160000*B^3CD^2+509598719/115200000*A^2C^2D^2+2267249639/518400000*ABC^2D^2+1959634247/691200000*B^2C^2D^2+9062329117/3110400000*AC^3D^2+13354106317/6220800000*BC^3D^2+892589/663552*C^4D^2+4215630661/1244160000*A^3D^3+6643266071/1555200000*A^2BD^3+4377980477/1244160000*AB^2D^3+1245650479/622080000*B^3D^3+5520812029/1555200000*A^2CD^3+114293357/32400000*ABCD^3+3541738373/1555200000*B^2CD^3+2730290939/1036800000*AC^2D^3+447457201/230400000*BC^2D^3+900095/663552*C^3D^3+131691713/55296000*A^2D^4+360449467/153600000*ABD^4+1050359257/691200000*B^2D^4+267623371/138240000*ACD^4+5936446979/4147200000*BCD^4+1449419/1327104*C^2D^4+15576417979/12441600000*AD^5+28824203411/31104000000*BD^5+990577/1327104*CD^5+4975/10368*D^6+2219663977/829440000*A^5E+5089795147/1036800000*A^4BE+38120496467/6220800000*A^3B^2E+16534511591/3110400000*A^2B^3E+13568188757/4147200000*AB^4E+607610891/414720000*B^5E+12771601327/3110400000*A^4CE+10113889817/1555200000*A^3BCE+2350239701/345600000*A^2B^2CE+7406093629/1555200000*AB^3CE+3627060493/1555200000*B^4CE+1820869331/388800000*A^3C^2E+6223530461/1036800000*A^2BC^2E+257637563/51840000*AB^2C^2E+17326115551/6220800000*B^3C^2E+11948955703/3110400000*A^2C^3E+61502281/16200000*ABC^3E+1912136987/777600000*B^2C^3E+13998635627/6220800000*AC^4E+20670772529/12441600000*BC^4E+414739/442368*C^5E+5512166149/1555200000*A^4DE+4415708561/777600000*A^3BDE+1544055233/259200000*A^2B^2DE+6477049373/1555200000*AB^3DE+3137374283/1555200000*B^4DE+68276569/14400000*A^3CDE+67550161/10800000*A^2BCDE+449870299/86400000*AB^2CDE+491419367/172800000*B^3CDE+7053264941/1555200000*A^2C^2DE+394447957/86400000*ABC^2DE+4551647707/1555200000*B^2C^2DE+4619906053/1555200000*AC^3DE+3417735437/1555200000*BC^3DE+12239/9216*C^4DE+145621697/38880000*A^3D^2E+3795976241/777600000*A^2BD^2E+12597544609/3110400000*AB^2D^2E+13912386961/6220800000*B^3D^2E+1403279729/345600000*A^2CD^2E+177866423/43200000*ABCD^2E+907936651/345600000*B^2CD^2E+9465059537/3110400000*AC^2D^2E+1752151279/777600000*BC^2D^2E+112487/73728*C^3D^2E+121680739/41472000*A^2D^3E+23858657/8100000*ABD^3E+217462259/115200000*B^2D^3E+945269303/388800000*ACD^3E+175323277/97200000*BCD^3E+37697/27648*C^2D^3E+1029416329/622080000*AD^4E+15268468459/12441600000*BD^4E+437363/442368*CD^4E+6803/10368*D^5E+6225617249/2073600000*A^4E^2+29330647081/6220800000*A^3BE^2+676871491/138240000*A^2B^2E^2+21391220633/6220800000*AB^3E^2+1762844143/1036800000*B^4E^2+2448264757/622080000*A^3CE^2+106014829/20736000*A^2BCE^2+2198860871/518400000*AB^2CE^2+14611710587/6220800000*B^3CE^2+771689789/207360000*A^2C^2E^2+42748589/11520000*ABC^2E^2+198371479/82944000*B^2C^2E^2+7608182243/3110400000*AC^3E^2+11237145071/6220800000*BC^3E^2+246695/221184*C^4E^2+329706661/97200000*A^3DE^2+27072383/6075000*A^2BDE^2+2302860373/622080000*AB^2DE^2+12626950097/6220800000*B^3DE^2+11535610697/3110400000*A^2CDE^2+32660341/8640000*ABCDE^2+59820527/24883200*B^2CDE^2+115459999/41472000*AC^2DE^2+535022747/259200000*BC^2DE^2+102247/73728*C^3DE^2+614763817/207360000*A^2D^2E^2+3108939047/1036800000*ABD^2E^2+158764979/82944000*B^2D^2E^2+7699692607/3110400000*ACD^2E^2+357505049/194400000*BCD^2E^2+11349/8192*C^2D^2E^2+63882301/34560000*AD^3E^2+2843122687/2073600000*BD^3E^2+81407/73728*CD^3E^2+8251/10368*D^4E^2+5586243353/2073600000*A^3E^3+1785917353/518400000*A^2BE^3+393052667/138240000*AB^2E^3+51759503/32400000*B^3E^3+4458406439/1555200000*A^2CE^3+93228749/32400000*ABCE^3+2869564831/1555200000*B^2CE^3+6630189199/3110400000*AC^2E^3+9805289773/6220800000*BC^2E^3+239957/221184*C^3E^3+7670191373/3110400000*A^2DE^3+3374077/1350000*ABDE^3+4951990597/3110400000*B^2DE^3+53520931/25920000*ACDE^3+49738403/32400000*BCDE^3+31867/27648*C^2DE^3+523414373/311040000*AD^2E^3+7772061653/6220800000*BD^2E^3+74215/73728*CD^2E^3+8251/10368*D^3E^3+7655288777/4147200000*A^2E^4+7633209847/4147200000*ABE^4+1225858907/1036800000*B^2E^4+9459687953/6220800000*ACE^4+2805758227/2488320000*BCE^4+376937/442368*C^2E^4+4051069607/3110400000*ADE^4+2409865727/2488320000*BDE^4+345691/442368*CDE^4+6803/10368*D^2E^4+19650443471/20736000000*AE^5+3647767961/5184000000*BE^5+251251/442368*CE^5+4975/10368*DE^5+45/128*E^6
        +711656152063/41489280000*A^5+6829064182987/207446400000*A^4B+59942258647/1440600000*A^3B^2+833289012173/23049600000*A^2B^3+9144107562571/414892800000*AB^4+816039511/86400000*B^5+2156647994093/77792400000*A^4C+11418701671397/248935680000*A^3BC+4020728108203/82978560000*A^2B^2C+1050130845899/31116960000*AB^3C+16447666823/1036800000*B^4C+19929115037941/622339200000*A^3C^2+533927048257/12446784000*A^2BC^2+1775726111987/49787136000*AB^2C^2+9969370201/518400000*B^3C^2+32694101200393/1244678400000*A^2C^3+249045556009/9219840000*ABC^3+1102773553/64800000*B^2C^3+4209071201131/276595200000*AC^4+72757461/6400000*BC^4+1048975963/172800000*C^5+665699976677/27659520000*A^4D+60107660374819/1493614080000*A^3BD+2653594736063/62233920000*A^2B^2D+8857974420247/298722816000*AB^3D+119235011/8640000*B^4D+2020804570241/59744563200*A^3CD+11566871550589/248935680000*A^2BCD+19330619482949/497871360000*AB^2CD+6368768609/311040000*B^3CD+6101324620049/186701760000*A^2C^2D+5667058235521/165957120000*ABC^2D+41471017/1944000*B^2C^2D+31758760360633/1493614080000*AC^3D+247453871/15552000*BC^3D+4736493041/518400000*C^4D+25053894319/972405000*A^3D^2+2094355551661/59744563200*A^2BD^2+1744851114791/59744563200*AB^2D^2+161066339/10368000*B^3D^2+43838777569777/1493614080000*A^2CD^2+1537256950499/49787136000*ABCD^2+11669063/607500*B^2CD^2+33001267133461/1493614080000*AC^2D^2+2573790233/155520000*BC^2D^2+2767798957/259200000*C^3D^2+3345157163669/165957120000*A^2D^3+654558854399/31116960000*ABD^3+272166167/20736000*B^2D^3+4351590604657/248935680000*ACD^3+680002267/51840000*BCD^3+9920879647/1036800000*C^2D^3+1242339192599/110638080000*AD^4+349402361/41472000*BD^4+14196441371/2073600000*CD^4+1414511/331776*D^5+2950789623857/138297600000*A^4E+268451736126523/7468070400000*A^3BE+47508510494783/1244678400000*A^2B^2E+198023140798991/7468070400000*AB^3E+1411491683/115200000*B^4E+7527398652353/248935680000*A^3CE+4335429062599/103723200000*A^2BCE+2718975172649/77792400000*AB^2CE+2112061469/115200000*B^3CE+10933878261791/373403520000*A^2C^2E+3190142769127/103723200000*ABC^2E+29778688639/1555200000*B^2C^2E+263189103629/13829760000*AC^3E+14787592687/1036800000*BC^3E+8419752257/1036800000*C^4E+39170318249917/1493614080000*A^3DE+45579414376171/1244678400000*A^2BDE+76318223834731/2489356800000*AB^2DE+49586487391/3110400000*B^3DE+19110279779047/622339200000*A^2CDE+22719049553/691488000*ABCDE+10491287737/518400000*B^2CDE+3219179068393/138297600000*AC^2DE+6050776459/345600000*BC^2DE+5713093603/518400000*C^3DE+8778505936447/373403520000*A^2D^2E+20758066246147/829785600000*ABD^2E+24016996073/1555200000*B^2D^2E+25908223505957/1244678400000*ACD^2E+16263596447/1036800000*BCD^2E+5868376633/518400000*C^2D^2E+21641188557161/1493614080000*AD^3E+33922543277/3110400000*BD^3E+287470631/32400000*CD^3E+323029/55296*D^4E+4435491862717/207446400000*A^3E^2+219762184328503/7468070400000*A^2BE^2+183365317567529/7468070400000*AB^2E^2+2234473073/172800000*B^3E^2+18422414707799/746807040000*A^2CE^2+8141684951341/311169600000*ABCE^2+50357989843/3110400000*B^2CE^2+578942727527/31116960000*AC^2E^2+4827504731/345600000*BC^2E^2+4617823189/518400000*C^3E^2+3542028319393/165957120000*A^2DE^2+3158584883777/138297600000*ABDE^2+4855780139/345600000*B^2DE^2+11834344542931/622339200000*ACDE^2+7439019091/518400000*BCDE^2+1782301661/172800000*C^2DE^2+22128262833569/1493614080000*AD^2E^2+34734010703/3110400000*BD^2E^2+4711839419/518400000*CD^2E^2+60893/9216*D^3E^2+13332027140147/829785600000*A^2E^3+10548644730083/622339200000*ABE^3+10888454717/1036800000*B^2E^3+3511560890917/248935680000*ACE^3+5502541051/518400000*BCE^3+994923559/129600000*C^2E^3+336056606831/27659520000*ADE^3+528034213/57600000*BDE^3+35849387/4800000*CDE^3+36715/6144*D^2E^3+14390998392191/1659571200000*AE^4+13534482521/2073600000*BE^4+344430653/64800000*CE^4+499807/110592*DE^4+8225/2592*E^5+1992303854291/103723200000*A^4F+10126871826971/311169600000*A^3BF+598364482717/17287200000*A^2B^2F+29904912385813/1244678400000*AB^3F+477040781/43200000*B^4F+204585281857703/7468070400000*A^3CF+47358630263837/1244678400000*A^2BCF+79246710827057/2489356800000*AB^2CF+51725881291/3110400000*B^3CF+172267718129/6482700000*A^2C^2F+23254496441749/829785600000*ABC^2F+375848669/21600000*B^2C^2F+128866772939041/7468070400000*AC^3F+40273709357/3110400000*BC^3F+632889131/86400000*C^4F+35505722193593/1493614080000*A^3DF+124525133622377/3734035200000*A^2BDF+208618625424347/7468070400000*AB^2DF+45001690979/3110400000*B^3DF+52241795395019/1867017600000*A^2CDF+10388501909/345744000*ABCDF+14357661577/777600000*B^2CDF+39627375353177/1867017600000*AC^2DF+49710435229/3110400000*BC^2DF+5189518663/518400000*C^3DF+10628805452837/497871360000*A^2D^2F+18923943534733/829785600000*ABD^2F+7279509769/518400000*B^2D^2F+141802191569393/7468070400000*ACD^2F+22277900057/1555200000*BCD^2F+10682656061/1036800000*C^2D^2F+9817128569471/746807040000*AD^3F+7703318917/777600000*BD^3F+4181415091/518400000*CD^3F+874699/165888*D^4F+436970655139/20744640000*A^3EF+6946201675321/233377200000*A^2BEF+93176329417373/3734035200000*AB^2EF+2219541919/172800000*B^3EF+2916687798727/116688600000*A^2CEF+46648021087/1728720000*ABCEF+12849209261/777600000*B^2CEF+17717918866583/933508800000*AC^2EF+22261977451/1555200000*BC^2EF+4613573713/518400000*C^3EF+168440482367/7779240000*A^2DEF+212293033/9003750*ABDEF+1488598181/103680000*B^2DEF+204020451317/10372320000*ACDEF+6440959/432000*BCDEF+1830169187/172800000*C^2DEF+11319170751361/746807040000*AD^2EF+1784825623/155520000*BD^2EF+4855265723/518400000*CD^2EF+61103/9216*D^3EF+2937178086763/165957120000*A^2E^2F+3959372951231/207446400000*ABE^2F+4036878013/345600000*B^2E^2F+59417839720427/3734035200000*ACE^2F+18716501461/1555200000*BCE^2F+4453760843/518400000*C^2E^2F+20491255685167/1493614080000*ADE^2F+1618172801/155520000*BDE^2F+8812662971/1036800000*CDE^2F+13755/2048*D^2E^2F+192253010309/18439680000*AE^3F+1021549921/129600000*BE^3F+2222759819/345600000*CE^3F+8407/1536*DE^3F+20755/5184*E^4F+313834311389/17287200000*A^3F^2+7843865847521/311169600000*A^2BF^2+26209999118437/1244678400000*AB^2F^2+79238369/7200000*B^3F^2+157962899188883/7468070400000*A^2CF^2+3122836778279/138297600000*ABCF^2+43278012211/3110400000*B^2CF^2+119317971533389/7468070400000*AC^2F^2+37379610653/3110400000*BC^2F^2+109359509/14400000*C^3F^2+13677526314799/746807040000*A^2DF^2+24548375871497/1244678400000*ABDF^2+37585874039/3110400000*B^2DF^2+40916905870201/2489356800000*ACDF^2+12882335963/1036800000*BCDF^2+9212115731/1036800000*C^2DF^2+19022222869423/1493614080000*AD^2F^2+7478362003/777600000*BD^2F^2+8125977083/1036800000*CD^2F^2+155953/27648*D^3F^2+537793708597/33191424000*A^2EF^2+21854003824501/1244678400000*ABEF^2+137137819/12800000*B^2EF^2+18230465280799/1244678400000*ACEF^2+718682839/64800000*BCEF^2+511287791/64800000*C^2EF^2+6289714576549/497871360000*ADEF^2+994568491/103680000*BDEF^2+2710255247/345600000*CDEF^2+37933/6144*D^2EF^2+96799369111/9219840000*AE^2F^2+915996823/115200000*BE^2F^2+2243857511/345600000*CE^2F^2+11319/2048*DE^2F^2+7595/1728*E^3F^2+5464551188069/414892800000*A^2F^3+5813426708531/414892800000*ABF^3+2984234323/345600000*B^2F^3+1816396597697/155584800000*ACF^3+2281999531/259200000*BCF^3+2186082689/345600000*C^2F^3+61861982827/6146560000*ADF^3+7890369851/1036800000*BDF^3+1609426903/259200000*CDF^3+818125/165888*D^2F^3+294661677727/33191424000*AEF^3+871687691/129600000*BEF^3+5697304487/1036800000*CEF^3+7189/1536*DEF^3+20755/5184*E^2F^3+5744500309657/829785600000*AF^4+3610635119/691200000*BF^4+2945172517/691200000*CF^4+1203461/331776*DF^4+8225/2592*EF^4+315/128*F^5
        +176797363471/1229312000*A^4+62124531251/245862400*A^3B+66664562929/245862400*A^2B^2+138187061627/737587200*AB^3+4797273331/57624000*B^4+7101326503189/33191424000*A^3C+1702191463939/5531904000*A^2BC+317572459411/1229312000*AB^2C+271747914023/2074464000*B^3C+6951753665719/33191424000*A^2C^2+841058732057/3687936000*ABC^2+287497168061/2074464000*B^2C^2+1496220226789/11063808000*AC^3+14177323409/138297600*BC^3+19987751/360000*C^4+6192813009221/33191424000*A^3D+35669858617/131712000*A^2BD+119922900169/526848000*AB^2D+475159685549/4148928000*B^3D+1264293773/5531904*A^2CD+466260128501/1843968000*ABCD+1685176463/11063808*B^2CD+128349541487/737587200*AC^2D+61001626397/460992000*BC^2D+45948289/576000*C^3D+5614765703399/33191424000*A^2D^2+6547426559/35123200*ABD^2+466452850139/4148928000*B^2D^2+575812656967/3687936000*ACD^2+4569522399/38416000*BCD^2+47900749/576000*C^2D^2+1144914856613/11063808000*AD^3+108938265539/1382976000*BD^3+4135533/64000*CD^3+6422659/160000*D^4+9185037560207/55319040000*A^3E+838756606613/3457440000*A^2BE+11288931001049/55319040000*AB^2E+706034638643/6914880000*B^3E+1700213182231/8297856000*A^2CE+6300989978627/27659520000*ABCE+2837773171193/20744640000*B^2CE+1036685266823/6638284800*AC^2E+2467284046529/20744640000*BC^2E+205098217/2880000*C^3E+2959935675499/16595712000*A^2DE+5528430857239/27659520000*ABDE+619333867009/5186160000*B^2DE+1546209676757/9219840000*ACDE+147873622957/1152480000*BCDE+42607811/480000*C^2DE+4161348893609/33191424000*AD^2E+1988289332653/20744640000*BD^2E+226901363/2880000*CD^2E+12802643/240000*D^3E+7794632324897/55319040000*A^2E^2+962795570381/6146560000*ABE^2+649673988353/6914880000*B^2E^2+4360297375939/33191424000*ACE^2+520278113231/5186160000*BCE^2+200614597/2880000*C^2E^2+755611745953/6638284800*ADE^2+1807933633423/20744640000*BDE^2+206515763/2880000*CDE^2+19733137/360000*D^2E^2+4558303024909/55319040000*AE^3+145020015517/2304960000*BE^3+148948469/2880000*CE^3+31876873/720000*DE^3+23495/768*E^4+395038755593/2634240000*A^3F+6090548066299/27659520000*A^2BF+10253012471069/55319040000*AB^2F+35475834943/384160000*B^3F+15442421566541/82978560000*A^2CF+638286510447/3073280000*ABCF+2580971648719/20744640000*B^2CF+23557834524991/165957120000*AC^2F+2245324923367/20744640000*BC^2F+10316083/160000*C^3F+2689814181413/16595712000*A^2DF+7386313391/40516875*ABDF+1127144257597/10372320000*B^2DF+8099734409/52920000*ACDF+151990783799/1296540000*BCDF+34929041/432000*C^2DF+3785062100563/33191424000*AD^2F+1810586141897/20744640000*BD^2F+62036813/864000*CD^2F+8700401/180000*D^3F+797061294157/5531904000*A^2EF+482982435653/2963520000*ABEF+223132504043/2304960000*B^2EF+355137651223/2593080000*ACEF+2179524479827/20744640000*BCEF+311629681/4320000*C^2EF+1972020410389/16595712000*ADEF+379190052863/4148928000*BDEF+13571771/180000*CDEF+41020673/720000*D^2EF+1047015837541/11063808000*AE^2F+167410578059/2304960000*BE^2F+517211071/8640000*CE^2F+73903013/1440000*DE^2F+87445/2304*E^3F+316242738173/2634240000*A^2F^2+2477601727313/18439680000*ABF^2+92490726449/1152480000*B^2F^2+2080029223729/18439680000*ACF^2+99464626669/1152480000*BCF^2+28612589/480000*C^2F^2+3246742149149/33191424000*ADF^2+1556567573737/20744640000*BDF^2+26704259/432000*CDF^2+33820649/720000*D^2F^2+959176326271/11063808000*AEF^2+25594949439/384160000*BEF^2+474841721/8640000*CEF^2+22629221/480000*DEF^2+88615/2304*E^2F^2+1601312192579/23708160000*AF^3+536202950737/10372320000*BF^3+20430013/480000*CF^3+78793969/2160000*DF^3+221725/6912*EF^3+3485/144*F^4+1515615517289/11063808000*A^3G+620572823923/3073280000*A^2BG+1045193338213/6146560000*AB^2G+36496894993/432180000*B^3G+4722810945343/27659520000*A^2CG+352432027673/1843968000*ABCG+790208352461/6914880000*B^2CG+2403040818071/18439680000*AC^2G+229253719391/2304960000*BC^2G+4720711/80000*C^3G+164597158123/1106380800*A^2DG+24867362269/148176000*ABDG+13809435419/138297600*B^2DG+146220488311/1037232000*ACDG+56045140717/518616000*BCDG+160605263/2160000*C^2DG+1158921712993/11063808000*AD^2G+110977056919/1382976000*BD^2G+285370763/4320000*CD^2G+31871419/720000*D^3G+731876066797/5531904000*A^2EG+622657051619/4148928000*ABEG+123061111441/1382976000*B^2EG+8725172047/69148800*ACEG+133984254077/1382976000*BCEG+286673207/4320000*C^2EG+1817594551141/16595712000*ADEG+69958411051/829785600*BDEG+2505427/36000*CDEG+37759103/720000*D^2EG+192435495409/2212761600*AE^2G+92391512963/1382976000*BE^2G+476042777/8640000*CE^2G+68054243/1440000*DE^2G+80125/2304*E^3G+94376711923/790272000*A^2FG+3762750990527/27659520000*ABFG+278143375613/3457440000*B^2FG+9497561823229/82978560000*ACFG+228163704421/2593080000*BCFG+28830457/480000*C^2FG+1649687247973/16595712000*ADFG+79465437371/1037232000*BDFG+22786573/360000*CDFG+34207403/720000*D^2FG+162629585821/1843968000*AEFG+15696711959/230496000*BEFG+6760157/120000*CEFG+726241/15000*DEFG+44945/1152*E^2FG+38945442749/526848000*AF^2G+43722736637/768320000*BF^2G+45121097/960000*CF^2G+6457377/160000*DF^2G+81865/2304*EF^2G+1355/48*F^3G+1151493594509/11063808000*A^2G^2+719845320603/6146560000*ABG^2+15065523539/216090000*B^2G^2+1814536438321/18439680000*ACG^2+86902359833/1152480000*BCG^2+12446633/240000*C^2G^2+2834217779581/33191424000*ADG^2+272169529369/4148928000*BDG^2+116853539/2160000*CDG^2+14728927/360000*D^2G^2+33510411671/442552320*AEG^2+161196677/2765952*BEG^2+415779067/8640000*CEG^2+59489173/1440000*DEG^2+77255/2304*E^2G^2+35933937773/526848000*AFG^2+121167247751/2304960000*BFG^2+41711257/960000*CFG^2+53751673/1440000*DFG^2+75745/2304*EFG^2+1355/48*F^2G^2+1878715888769/33191424000*AG^3+14069258887/324135000*BG^3+25763903/720000*CG^3+16576691/540000*DG^3+186725/6912*EG^3+3485/144*FG^3+315/16*G^4
        +10647273349/9031680*A^3+7068227681/3951360*A^2B+95530136083/63221760*AB^2+5144965209/7024640*B^3+1716506341/1128960*A^2C+5519378419/3161088*ABC+450800681/439040*B^2C+328691099/282240*AC^2+6298442019/7024640*BC^2+113040321/219520*C^3+52574179337/39513600*A^2D+243235243247/158054400*ABD+494475419/548800*B^2D+102695607457/79027200*ACD+17641686093/17561600*BCD+2966312667/4390400*C^2D+297363394073/316108800*AD^2+25529238639/35123200*BD^2+82626519/137200*CD^2+2484903/6400*D^3+469309721387/395136000*A^2E+5567344843/4032000*ABE+35381860191/43904000*B^2E+92221994863/79027200*ACE+39665488301/43904000*BCE+26569659699/43904000*C^2E+160798777631/158054400*ADE+17332419611/21952000*BDE+28793318871/43904000*CDE+1924077/4000*D^2E+619895540267/790272000*AE^2+106718762547/175616000*BE^2+11072413251/21952000*CE^2+27832281/64000*DE^2+3921507/12800*E^3+121439702263/112896000*A^2F+70862331719/56448000*ABF+16373563/22400*B^2F+104875844147/98784000*ACF+7225627917/8780800*BCF+24125688309/43904000*C^2F+146368431131/158054400*ADF+4512814829/6272000*BDF+26259994903/43904000*CDF+13988757/32000*D^2F+130353411149/158054400*AEF+56364241663/87808000*BEF+2345551337/4390400*CEF+14765583/32000*DEF+1154361/3200*E^2F+151133301883/225792000*AF^2+372448203/716800*BF^2+4740247353/10976000*CF^2+23855147/64000*DF^2+4218437/12800*EF^2+127815/512*F^3+4865189509/4939200*A^2G+227797155049/197568000*ABG+58843541983/87808000*B^2G+385508840777/395136000*ACG+4747297197/6272000*BCG+553431759/1097600*C^2G+67282900639/79027200*ADG+2906865631/4390400*BDG+1511236511/2744000*CDG+12844563/32000*D^2G+7492833641/9878400*AEG+5188399257/8780800*BEG+21604738087/43904000*CEG+13607109/32000*DEG+132567/400*E^2G+553132427/806400*AFG+5872558579/10976000*BFG+19580709339/43904000*CFG+1234089/3200*DFG+546303/1600*EFG+8985/32*F^2G+45937136327/79027200*AG^2+79372967613/175616000*BG^2+825565971/2195200*CG^2+2598791/8000*DG^2+1839431/6400*EG^2+132405/512*FG^2+13335/64*G^3+143811767681/158054400*A^2H+16874473031/15805440*ABH+10881345627/17561600*B^2H+35712373691/39513600*ACH+6161487993/8780800*BCH+1280151657/2744000*C^2H+12470233793/15805440*ADH+5391571233/8780800*BDH+2243581413/4390400*CDH+5945487/16000*D^2H+5556553111/7902720*AEH+24065143/43904*BEH+4010419923/8780800*CEH+2526861/6400*DEH+982251/3200*E^2H+14360409809/22579200*AFH+10898051131/21952000*BFH+18177874941/43904000*CFH+11461301/32000*DFH+2030117/6400*EFH+8325/32*F^2H+164214403/282240*AGH+39919338861/87808000*BGH+2082276471/5488000*CGH+5254451/16000*DGH+931121/3200*EGH+67095/256*FGH+14385/64*G^2H+161798287211/316108800*AH^2+13996895337/35123200*BH^2+1821448917/5488000*CH^2+4590111/16000*DH^2+3250647/12800*EH^2+117045/512*FH^2+13335/64*GH^2+2835/16*H^3
        +71569793/8064*A^2+57186331/5376*AB+196511005/32256*B^2+1021397467/112896*AC+199725437/28224*BC+115809405/25088*C^2+255768271/32256*AD+701763029/112896*BD+130513875/25088*CD+11591155/3136*D^2+2668366121/376320*AE+785690323/141120*BE+585143823/125440*CE+63513663/15680*DE+196289/64*E^2+259387489/40320*AF+407831203/80640*BF+75997737/17920*CF+23112373/6272*DF+26207/8*EF+334047/128*F^2+41637947/7056*AG+1310578181/282240*BG+488769731/125440*CG+106229087/31360*DG+48201/16*EG+87113/32*FG+289625/128*G^2+136974633/25088*AH+485398307/112896*BH+452809837/125440*CH+24613713/7840*DH+357509/128*EH+20197/8*FH+295575/128*GH+15925/8*H^2+164078297/32256*AI+56565055/14112*BI+84465945/25088*CI+4592981/1568*DI+333655/128*EI+150831/64*FI+275975/128*GI+15925/8*HI+14175/8*I^2
        +888057/16*A+704253/16*B+2377815/64*C+14539635/448*D+3245715/112*E+420453/16*F+771309/32*G+356895/16*H+20790*I+155925/8*J
        467775/2
       
1/11264 6496405/2327947776 569783399443/29797731532800 62996436704843/969978240000000 161010876007913/1149603840000000 1315083796508393489/6210447344640000000 192607568465219557/828059645952000000 729523557898043/3943141171200000 4609294575701/45064470528000 42308191/1192181760 3/512
Average 865440371749038462403/111788052203520000000
Level 12:
        +1/2048*A^11
        +58537/20155392*A^10+2059/373248*A^9B+1771/186624*A^8B^2+11771/839808*A^7B^3+89/5184*A^6B^4+89/5184*A^5B^5+11771/839808*A^4B^6+1771/186624*A^3B^7+2059/373248*A^2B^8+58537/20155392*AB^9+3/2048*B^10
        +98214569/5159780352*A^9+33279497/859963392*A^8B+56802215/859963392*A^7B^2+79395397/859963392*A^6B^3+44585317/429981696*A^5B^4+80043307/859963392*A^4B^5+57951521/859963392*A^3B^6+34730503/859963392*A^2B^7+35927033/1719926784*AB^8+97391/10077696*B^9+82199711/2579890176*A^8C+839755/13436928*A^7BC+9701335/95551488*A^6B^2C+18777041/143327232*A^5B^3C+37888561/286654464*A^4B^4C+832111/7962624*A^3B^5C+18936131/286654464*A^2B^6C+14973619/429981696*AB^7C+164347/10077696*B^8C+127683673/2579890176*A^7C^2+25456633/286654464*A^6BC^2+3132047/23887872*A^5B^2C^2+7206163/47775744*A^4B^3C^2+12728531/95551488*A^3B^4C^2+8747479/95551488*A^2B^5C^2+7349641/143327232*AB^6C^2+257555/10077696*B^7C^2+170214179/2579890176*A^6C^3+45192655/429981696*A^5BC^3+19501477/143327232*A^4B^2C^3+539417/3981312*A^3B^3C^3+29626099/286654464*A^2B^4C^3+27195305/429981696*AB^5C^3+346819/10077696*B^6C^3+93122921/1289945088*A^5C^4+85989791/859963392*A^4BC^4+1174831/10616832*A^3B^2C^4+2995943/31850496*A^2B^3C^4+54669871/859963392*AB^4C^4+384611/10077696*B^5C^4+164382989/2579890176*A^4C^5+5408659/71663616*A^3BC^5+2243737/31850496*A^2B^2C^5+7482439/143327232*AB^3C^5+346819/10077696*B^4C^5+117339919/2579890176*A^3C^6+39155725/859963392*A^2BC^6+15574759/429981696*AB^2C^6+257555/10077696*B^3C^6+69140657/2579890176*A^2C^7+3237637/143327232*ABC^7+164347/10077696*B^2C^7+69514979/5159780352*AC^8+97391/10077696*BC^8+3/512*C^9
        +2066402017411/14929920000000*A^8+4367040753017/14929920000000*A^7B+7233436600721/14929920000000*A^6B^2+5627615105891/8957952000000*A^5B^3+5683432004561/8957952000000*A^4B^4+7484999097731/14929920000000*A^3B^5+14138544282401/44789760000000*A^2B^6+7319260234153/44789760000000*AB^7+40712485/573308928*B^8+16296726609839/67184640000000*A^7C+902089342699/1866240000000*A^6BC+277517194727/373248000000*A^5B^2C+54024708251/62208000000*A^4B^3C+15944208149/20736000000*A^3B^4C+975762265769/1866240000000*A^2B^5C+525796912829/1866240000000*AB^6C+216733675/1719926784*B^7C+8203115484119/22394880000000*A^6C^2+162725721193/248832000000*A^5BC^2+332829321259/373248000000*A^4B^2C^2+112548021263/124416000000*A^3B^3C^2+511224452419/746496000000*A^2B^4C^2+62191327001/155520000000*AB^5C^2+110598025/573308928*B^6C^2+6078860716349/13436928000000*A^5C^3+130838314603/186624000000*A^4BC^3+304946705689/373248000000*A^3B^2C^3+130784578573/186624000000*A^2B^3C^3+339388068383/746496000000*AB^4C^3+416597015/1719926784*B^5C^3+5964356405279/13436928000000*A^4C^4+145049071213/248832000000*A^3BC^4+140900840873/248832000000*A^2B^2C^4+25564896359/62208000000*AB^3C^4+418605005/1719926784*B^4C^4+7690808654909/22394880000000*A^3C^5+699196939819/1866240000000*A^2BC^5+1126237411549/3732480000000*AB^2C^5+112472395/573308928*B^3C^5+14220874518239/67184640000000*A^2C^6+117592183193/622080000000*ABC^6+225099625/1719926784*B^2C^6+7136239317467/67184640000000*AC^7+132537565/1719926784*BC^7+97135/2239488*C^8+27950374014323/134369280000000*A^7D+1044861291437/2488320000000*A^6BD+1937091829931/2985984000000*A^5B^2D+566499144517/746496000000*A^4B^3D+668918487557/995328000000*A^3B^4D+3407825195791/7464960000000*A^2B^5D+3657719729137/14929920000000*AB^6D+186000035/1719926784*B^7D+1298200889593/3732480000000*A^6CD+26828290721/41472000000*A^5BCD+28091320049/31104000000*A^4B^2CD+57509316883/62208000000*A^3B^3CD+43385352509/62208000000*A^2B^4CD+3445175599/8640000000*AB^5CD+17582705/95551488*B^6CD+732607296953/1492992000000*A^5C^2D+37105485011/46656000000*A^4BC^2D+4947444001/5184000000*A^3B^2C^2D+948602567/1152000000*A^2B^3C^2D+24369637549/46656000000*AB^4C^2D+50577665/191102976*B^5C^2D+612342972613/1119744000000*A^4C^3D+69626415637/93312000000*A^3BC^3D+961224241/1296000000*A^2B^2C^3D+99408437803/186624000000*AB^3C^3D+43345405/143327232*B^4C^3D+701058736573/1492992000000*A^3C^4D+97971943027/186624000000*A^2BC^4D+39727869823/93312000000*AB^2C^4D+51648685/191102976*B^3C^4D+1159492834333/3732480000000*A^2C^5D+57852518251/207360000000*ABC^5D+18417185/95551488*B^2C^5D+3609232324393/22394880000000*AC^6D+66886325/573308928*BC^6D+149305/2239488*C^7D+13001872533353/44789760000000*A^6D^2+1579226671771/2985984000000*A^5BD^2+1084980426341/1492992000000*A^4B^2D^2+367806996137/497664000000*A^3B^3D^2+1668936812981/2985984000000*A^2B^4D^2+4839702923551/14929920000000*AB^5D^2+87799625/573308928*B^6D^2+652985798123/1492992000000*A^5CD^2+66958457027/93312000000*A^4BCD^2+35906894053/41472000000*A^3B^2CD^2+3876225599/5184000000*A^2B^3CD^2+352588939769/746496000000*AB^4CD^2+45155225/191102976*B^5CD^2+136511248261/248832000000*A^4C^2D^2+5259099563/6912000000*A^3BC^2D^2+439937429/576000000*A^2B^2C^2D^2+7550738497/13824000000*AB^3C^2D^2+9701365/31850496*B^4C^2D^2+396268736393/746496000000*A^3C^3D^2+18623508619/31104000000*A^2BC^3D^2+15154975037/31104000000*AB^2C^3D^2+29296865/95551488*B^3C^3D^2+578562241703/1492992000000*A^2C^4D^2+86403103723/248832000000*ABC^4D^2+45970895/191102976*B^2C^4D^2+538103284571/2488320000000*AC^5D^2+40925/262144*BC^5D^2+71215/746496*C^6D^2+9245689364483/26873856000000*A^5D^3+1221773150431/2239488000000*A^4BD^3+957486154451/1492992000000*A^3B^2D^3+411199915157/746496000000*A^2B^3D^3+3179540710991/8957952000000*AB^4D^3+317632495/1719926784*B^5D^3+18692745739/41472000000*A^4CD^3+116900670019/186624000000*A^3BCD^3+6502510093/10368000000*A^2B^2CD^3+20939825417/46656000000*AB^3CD^3+1327495/5308416*B^4CD^3+360323921113/746496000000*A^3C^2D^3+17051474579/31104000000*A^2BC^2D^3+6946821221/15552000000*AB^2C^2D^3+26685385/95551488*B^3C^2D^3+5426988733/13824000000*A^2C^3D^3+21895970431/62208000000*ABC^3D^3+3883565/15925248*B^2C^3D^3+1079982733733/4478976000000*AC^4D^3+99666365/573308928*BC^4D^3+260885/2239488*C^5D^3+8857813128713/26873856000000*A^4D^4+1322824427791/2985984000000*A^3BD^4+431544004087/995328000000*A^2B^2D^4+935835838577/2985984000000*AB^3D^4+311885005/1719926784*B^4D^4+181991682361/497664000000*A^3CD^4+19440049673/46656000000*A^2BCD^4+253032582589/746496000000*AB^2CD^4+13465975/63700992*B^3CD^4+484410888943/1492992000000*A^2C^2D^4+72812615063/248832000000*ABC^2D^4+38577295/191102976*B^2C^2D^4+327161492351/1492992000000*AC^3D^4+30195815/191102976*BC^3D^4+260885/2239488*C^4D^4+11232062395043/44789760000000*A^3D^5+695573084117/2488320000000*A^2BD^5+1123334081257/4976640000000*AB^2D^5+82468355/573308928*B^3D^5+286487383171/1244160000000*A^2CD^5+21742813543/103680000000*ABCD^5+1523545/10616832*B^2CD^5+1242442364273/7464960000000*AC^2D^5+22981405/191102976*BC^2D^5+71215/746496*C^3D^5+20439252030473/134369280000000*A^2D^6+2062778422037/14929920000000*ABD^6+162522665/1719926784*B^2D^6+2530890875243/22394880000000*ACD^6+47029165/573308928*BCD^6+149305/2239488*C^2D^6+10035284471429/134369280000000*AD^7+93572645/1719926784*BD^7+97135/2239488*CD^7+15/512*D^8
        +76184538889/69120000000*A^7+732293254079/311040000000*A^6B+229722450517/62208000000*A^5B^2+16864450321/3888000000*A^4B^3+239070960107/62208000000*A^3B^4+809583600799/311040000000*A^2B^5+857074853573/622080000000*AB^6+238378239881/414720000000*B^7+229278212087/116640000000*A^6C+720281128589/186624000000*A^5BC+56646429871/10368000000*A^4B^2C+174530002013/31104000000*A^3B^3C+262904602643/62208000000*A^2B^4C+1114003162627/466560000000*AB^5C+1948958934517/1866240000000*B^6C+525438315053/186624000000*A^5C^2+225230481047/46656000000*A^4BC^2+548352162151/93312000000*A^3B^2C^2+236930190997/46656000000*A^2B^3C^2+594512200267/186624000000*AB^4C^2+285048939493/186624000000*B^5C^2+16301927419/5184000000*A^4C^3+106054752751/23328000000*A^3BC^3+17808591461/3888000000*A^2B^2C^3+304587127313/93312000000*AB^3C^3+18172748563/10368000000*B^4C^3+503168063153/186624000000*A^3C^4+297821590327/93312000000*A^2BC^4+121414662433/46656000000*AB^2C^4+146331066929/93312000000*B^3C^4+1656545781391/933120000000*A^2C^5+521282408893/311040000000*ABC^5+1041744724781/933120000000*B^2C^5+1693091530777/1866240000000*AC^6+1242292695257/1866240000000*BC^6+5655841/15925248*C^7+632579677127/373248000000*A^6D+628484374313/186624000000*A^5BD+893344521131/186624000000*A^4B^2D+153105661981/31104000000*A^3B^3D+1383053689721/373248000000*A^2B^4D+486628056797/233280000000*AB^5D+1682242798217/1866240000000*B^6D+1048144999849/373248000000*A^5CD+323209651/64800000*A^4BCD+46206877853/7464960000*A^3B^2CD+100183201/18662400*A^2B^3CD+9176417921/2764800000*AB^4CD+571989442817/373248000000*B^5CD+340464238367/93312000000*A^4C^2D+102056690941/18662400000*A^3BC^2D+388332211/69120000*A^2B^2C^2D+148149459907/37324800000*AB^3C^2D+382830165467/186624000000*B^4C^2D+166297164961/46656000000*A^3C^3D+8080405501/1866240000*A^2BC^3D+53074597/14929920*AB^2C^3D+195098749771/93312000000*B^3C^3D+482378156207/186624000000*A^2C^4D+2268188971/921600000*ABC^4D+609597510167/373248000000*B^2C^4D+2609309546209/1866240000000*AC^5D+1912175344217/1866240000000*BC^5D+249865/442368*C^6D+67091444371/29859840000*A^5D^2+183140931169/46656000000*A^4BD^2+1795715145083/373248000000*A^3B^2D^2+194177078909/46656000000*A^2B^3D^2+1938839117171/746496000000*AB^4D^2+456071621581/373248000000*B^5D^2+203242084613/62208000000*A^4CD^2+20526013711/4147200000*A^3BCD^2+21171788581/4147200000*A^2B^2CD^2+3732264379/1036800000*AB^3CD^2+114506453869/62208000000*B^4CD^2+227384396477/62208000000*A^3C^2D^2+1552019999/345600000*A^2BC^2D^2+3067664221/829440000*AB^2C^2D^2+267807785927/124416000000*B^3C^2D^2+279208882367/93312000000*A^2C^3D^2+35515046183/12441600000*ABC^3D^2+176745221311/93312000000*B^2C^3D^2+221299894129/124416000000*AC^4D^2+323775771439/248832000000*BC^4D^2+4130171/5308416*C^5D^2+7481203229/3110400000*A^4D^3+662735064121/186624000000*A^3BD^3+56019163823/15552000000*A^2B^2D^3+477413629187/186624000000*AB^3D^3+41847444479/31104000000*B^4D^3+550694941559/186624000000*A^3CD^3+33959529977/9331200000*A^2BCD^3+111802987363/37324800000*AB^2CD^3+162141718631/93312000000*B^3CD^3+126904235381/46656000000*A^2C^2D^3+6485347333/2488320000*ABC^2D^3+80463334213/46656000000*B^2C^2D^3+112375165841/62208000000*AC^3D^3+41089016369/31104000000*BC^3D^3+1741249/1990656*C^4D^3+299735982929/149299200000*A^3D^4+452674193663/186624000000*A^2BD^4+1480208983411/746496000000*AB^2D^4+437788147861/373248000000*B^3D^4+750335959489/373248000000*A^2CD^4+2680577741/1382400000*ABCD^4+476217446657/373248000000*B^2CD^4+275453297851/186624000000*AC^2D^4+806939066867/746496000000*BC^2D^4+4184009/5308416*C^3D^4+241376362837/186624000000*A^2D^5+772137901397/622080000000*ABD^5+47670669991/58320000000*B^2D^5+158853720377/155520000000*ACD^5+467025988639/622080000000*BCD^5+774671/1327104*C^2D^5+481085596373/746496000000*AD^6+886191956557/1866240000000*BD^6+6074707/15925248*CD^6+15053/62208*D^7+103263866209/69120000000*A^6E+279385898611/93312000000*A^5BE+796591381939/186624000000*A^4B^2E+22774462439/5184000000*A^3B^3E+1233904050649/373248000000*A^2B^4E+1732172469407/933120000000*AB^5E+18322371317/23040000000*B^6E+310897413539/124416000000*A^5CE+52146892231/11664000000*A^4BCE+1423579481/256000000*A^3B^2CE+2084071933/432000000*A^2B^3CE+1111366576123/373248000000*AB^4CE+169819051789/124416000000*B^5CE+33764009419/10368000000*A^4C^2E+152879560259/31104000000*A^3BC^2E+8742116473/1728000000*A^2B^2C^2E+222121470881/62208000000*AB^3C^2E+4223198147/2304000000*B^4C^2E+4125933553/1296000000*A^3C^3E+60549388763/15552000000*A^2BC^3E+49749828709/15552000000*AB^2C^3E+4847387551/2592000000*B^3C^3E+143532596057/62208000000*A^2C^4E+274894377161/124416000000*ABC^4E+181690891429/124416000000*B^2C^4E+258106690693/207360000000*AC^5E+568306123709/622080000000*BC^5E+661697/1327104*C^6E+20092103009/9331200000*A^5DE+90952272829/23328000000*A^4BDE+454257261119/93312000000*A^3B^2DE+49287624317/11664000000*A^2B^3DE+485320141663/186624000000*AB^4DE+439679121277/373248000000*B^5DE+25269413807/7776000000*A^4CDE+1971628303/388800000*A^3BCDE+1830766961/345600000*A^2B^2CDE+1443774653/388800000*AB^3CDE+1791166247/972000000*B^4CDE+6409520579/1728000000*A^3C^2DE+7266005663/1555200000*A^2BC^2DE+12022183763/3110400000*AB^2C^2DE+22801347887/10368000000*B^3C^2DE+142209192337/46656000000*A^2C^3DE+1150490371/388800000*ABC^3DE+45300063433/23328000000*B^2C^3DE+55596001547/31104000000*AC^4DE+9070146823/6912000000*BC^4DE+666455/884736*C^5DE+97044152053/37324800000*A^4D^2E+745785107123/186624000000*A^3BD^2E+85760468507/20736000000*A^2B^2D^2E+135812916589/46656000000*AB^3D^2E+273992839567/186624000000*B^4D^2E+25837376113/7776000000*A^3CD^2E+26286246253/6220800000*A^2BCD^2E+10889118343/3110400000*AB^2CD^2E+1919645773/972000000*B^3CD^2E+10825440383/3456000000*A^2C^2D^2E+1268526031/414720000*ABC^2D^2E+41500263709/20736000000*B^2C^2D^2E+385524933569/186624000000*AC^3D^2E+35358185129/23328000000*BC^3D^2E+637561/663552*C^4D^2E+10097171393/4147200000*A^3D^3E+94440075077/31104000000*A^2BD^3E+155575417787/62208000000*AB^2D^3E+7447446977/5184000000*B^3D^3E+241580437/96000000*A^2CD^3E+960682673/388800000*ABCD^3E+25032229337/15552000000*B^2CD^3E+29047457317/15552000000*AC^2D^3E+28443899383/20736000000*BC^2D^3E+1289675/1327104*C^3D^3E+127832889637/74649600000*A^2D^4E+5761880837/3456000000*ABD^4E+406625469967/373248000000*B^2D^4E+56886055601/41472000000*ACD^4E+125675887079/124416000000*BCD^4E+1036357/1327104*C^2D^4E+334748550037/373248000000*AD^5E+616913372351/933120000000*BD^5E+1408645/2654208*CD^5E+7123/20736*D^6E+51186241459/27648000000*A^5E^2+4250221301/1296000000*A^4BE^2+502649909479/124416000000*A^3B^2E^2+27198475481/7776000000*A^2B^3E^2+180347610061/82944000000*AB^4E^2+5229004183/5184000000*B^5E^2+510013936787/186624000000*A^4CE^2+781522171679/186624000000*A^3BCE^2+89876901061/20736000000*A^2B^2CE^2+142347806107/46656000000*AB^3CE^2+287927107627/186624000000*B^4CE^2+63736048957/20736000000*A^3C^2E^2+59342938277/15552000000*A^2BC^2E^2+195736160299/62208000000*AB^2C^2E^2+75242500459/41472000000*B^3C^2E^2+26103845999/10368000000*A^2C^3E^2+150697702511/62208000000*ABC^3E^2+1840816621/1152000000*B^2C^3E^2+556381372921/373248000000*AC^4E^2+815821161427/746496000000*BC^4E^2+379339/589824*C^5E^2+87787526609/37324800000*A^4DE^2+680765054263/186624000000*A^3BDE^2+78547416607/20736000000*A^2B^2DE^2+62095892797/23328000000*AB^3DE^2+248313375647/186624000000*B^4DE^2+11798556139/3888000000*A^3CDE^2+2685594973/691200000*A^2BCDE^2+668200961/207360000*AB^2CDE^2+28101113573/15552000000*B^3CDE^2+89305025167/31104000000*A^2C^2DE^2+5842193459/2073600000*ABC^2DE^2+114310261987/62208000000*B^2C^2DE^2+352871917609/186624000000*AC^3DE^2+64802344433/46656000000*BC^3DE^2+578555/663552*C^4DE^2+30467765939/12441600000*A^3D^2E^2+48111810589/15552000000*A^2BD^2E^2+159028783403/62208000000*AB^2D^2E^2+180564538807/124416000000*B^3D^2E^2+19936208453/7776000000*A^2CD^2E^2+1314185993/518400000*ABCD^2E^2+102343581047/62208000000*B^2CD^2E^2+39641683669/20736000000*AC^2D^2E^2+29144585777/20736000000*BC^2D^2E^2+290347/294912*C^3D^2E^2+2980514813/1555200000*A^2D^3E^2+116588772647/62208000000*ABD^3E^2+38025750347/31104000000*B^2D^3E^2+287616114299/186624000000*ACD^3E^2+26484334049/23328000000*BCD^3E^2+194087/221184*C^2D^3E^2+81602215771/74649600000*AD^4E^2+600926495597/746496000000*BD^4E^2+380687/589824*CD^4E^2+9307/20736*D^5E^2+4951441513/2592000000*A^4E^3+178220431513/62208000000*A^3BE^3+2839048877/972000000*A^2B^2E^3+128710066051/62208000000*AB^3E^3+33327592559/31104000000*B^4E^3+444845329367/186624000000*A^3CE^3+138879761269/46656000000*A^2BCE^3+457954446803/186624000000*AB^2CE^3+65677624543/46656000000*B^3CE^3+103063687313/46656000000*A^2C^2E^3+132970776361/62208000000*ABC^2E^3+4096643713/2916000000*B^2C^2E^3+273091899839/186624000000*AC^3E^3+50052386641/46656000000*BC^3E^3+463165/663552*C^4E^3+76492248101/37324800000*A^3DE^3+8948366933/3456000000*A^2BDE^3+132953265127/62208000000*AB^2DE^3+28306353619/23328000000*B^3DE^3+50098535371/23328000000*A^2CDE^3+55109659/25920000*ABCDE^3+64276059091/46656000000*B^2CDE^3+74579976961/46656000000*AC^2DE^3+219502182587/186624000000*BC^2DE^3+1091875/1327104*C^3DE^3+4088938343/2332800000*A^2D^2E^3+107068683407/62208000000*ABD^2E^3+104496270691/93312000000*B^2D^2E^3+88068177713/62208000000*ACD^2E^3+8116051493/7776000000*BCD^2E^3+177961/221184*C^2D^2E^3+20541946927/18662400000*AD^3E^3+151254106409/186624000000*BD^3E^3+287369/442368*CD^3E^3+30817/62208*D^4E^3+14366632423/9216000000*A^3E^4+356834215757/186624000000*A^2BE^4+1169136669709/746496000000*AB^2E^4+18949617637/20736000000*B^3E^4+197420508319/124416000000*A^2CE^4+96220162013/62208000000*ABCE^4+125711598959/124416000000*B^2CE^4+12108587851/10368000000*AC^2E^4+71133764033/82944000000*BC^2E^4+1089299/1769472*C^3E^4+101687755033/74649600000*A^2DE^4+1158693853/864000000*ABDE^4+324735490567/373248000000*B^2DE^4+137407566163/124416000000*ACDE^4+101423942419/124416000000*BCDE^4+829799/1327104*C^2DE^4+68540961539/74649600000*AD^2E^4+505324238917/746496000000*BD^2E^4+960901/1769472*CD^2E^4+9307/20736*D^3E^4+34102578077/34560000000*A^2E^5+596184503483/622080000000*ABE^5+16224234371/25920000000*B^2E^5+368496702833/466560000000*ACE^5+1086381855367/1866240000000*BCE^5+198263/442368*C^2E^5+50408931877/74649600000*ADE^5+465705576121/933120000000*BDE^5+1064981/2654208*CDE^5+7123/20736*D^2E^5+200017633259/414720000000*AE^6+110871137407/311040000000*BE^6+1523161/5308416*CE^6+15053/62208*DE^6+45/256*E^7
        +492373056010501/52276492800000*A^6+28659948705197/1452124800000*A^5B+6893629400249/242020800000*A^4B^2+8540986374293/290424960000*A^3B^3+5347887430153/242020800000*A^2B^4+713188321310113/58084992000000*AB^5+78274456771/15552000000*B^6+1733893259479531/104552985600000*A^5C+64811364095819/2091059712000*A^4BC+2038035397769023/52276492800000*A^3B^2C+1770360470350457/52276492800000*A^2B^3C+270850711928443/13069123200000*AB^4C+22658556011/2488320000*B^5C+1143874577063011/52276492800000*A^4C^2+451621441731583/13069123200000*A^3BC^2+2783456549443/77446656000*A^2B^2C^2+1318467288311351/52276492800000*AB^3C^2+77162848693/6220800000*B^4C^2+12444362935169/580849920000*A^3C^3+716766544646869/26138246400000*A^2BC^3+1182946607249389/52276492800000*AB^2C^3+658594699/51840000*B^3C^3+809085336388117/52276492800000*A^2C^4+268905036654251/17425497600000*ABC^4+61658961721/6220800000*B^2C^4+2875482016458137/348509952000000*AC^5+63514607989/10368000000*BC^5+32813752181/10368000000*C^6+2003434092865489/139403980800000*A^5D+8517871605708221/313658956800000*A^4BD+21506961835351177/627317913600000*A^3B^2D+9346247889966499/313658956800000*A^2B^3D+22815348480854833/1254635827200000*AB^4D+98331168103/12441600000*B^5D+7146387457461421/313658956800000*A^4CD+1937392471261223/52276492800000*A^3BCD+4084627060605533/104552985600000*A^2B^2CD+11396652934199/418211942400*AB^3CD+121380040621/9331200000*B^4CD+257730449330021/9801842400000*A^3C^2D+10836713525382517/313658956800000*A^2BC^2D+2249842615634729/78414739200000*AB^2C^2D+587157373579/37324800000*B^3C^2D+6775197878262913/313658956800000*A^2C^3D+35601148834669/1633640400000*ABC^3D+129873756649/9331200000*B^2C^3D+7867777118940337/627317913600000*AC^4D+695155850473/74649600000*BC^4D+126259704053/24883200000*C^5D+734946622882963/41821194240000*A^4D^2+5902975121680477/209105971200000*A^3BD^2+411460745488969/13940398080000*A^2B^2D^2+4319858541389921/209105971200000*AB^3D^2+2486938811/248832000*B^4D^2+823395806018183/34850995200000*A^3CD^2+39297271217171/1254635827200*A^2BCD^2+817005025918561/31365895680000*AB^2CD^2+6529327427/460800000*B^3CD^2+52277588668703/2323399680000*A^2C^2D^2+1193368217338663/52276492800000*ABC^2D^2+12059942741/829440000*B^2C^2D^2+1540619521128827/104552985600000*AC^3D^2+27211013837/2488320000*BC^3D^2+81825407209/12441600000*C^4D^2+688383593010079/41821194240000*A^3D^3+3365620237205671/156829478400000*A^2BD^3+11136237421382837/627317913600000*AB^2D^3+813251761/82944000*B^3D^3+2807606341654489/156829478400000*A^2CD^3+238568729936941/13069123200000*ABCD^3+108058304627/9331200000*B^2CD^3+4186603568132969/313658956800000*AC^2D^3+74016218773/7464960000*BC^2D^3+16624863353/2488320000*C^3D^3+321268605293333/27880796160000*A^2D^4+1626841013312267/139403980800000*ABD^4+410149523/55296000*B^2D^4+2021067411438557/209105971200000*ACD^4+59704257431/8294400000*BCD^4+133509912221/24883200000*C^2D^4+2491220010101719/418211942400000*AD^5+3684185869/829440000*BD^5+447405287077/124416000000*CD^5+8701133/3981312*D^6+1773481875340669/139403980800000*A^5E+7599754584463861/313658956800000*A^4BE+19239512307496463/627317913600000*A^3B^2E+8364180771113567/313658956800000*A^2B^3E+4075534101955039/250927165440000*AB^4E+174286155211/24883200000*B^5E+797682780800599/39207369600000*A^4CE+64502487686843/1936166400000*A^3BCE+1839248389126229/52276492800000*A^2B^2CE+5934042471449/242020800000*AB^3CE+43409195207/3732480000*B^4CE+2461575762534389/104552985600000*A^3C^2E+4881853993644913/156829478400000*A^2BC^2E+202849106860723/7841473920000*AB^2C^2E+175505628389/12441600000*B^3C^2E+758505758235191/39207369600000*A^2C^3E+1025744593791931/52276492800000*ABC^3E+233029188497/18662400000*B^2C^3E+7031772580111997/627317913600000*AC^4E+24889025567/2985984000*BC^4E+55993271159/12441600000*C^5E+1841991384779593/104552985600000*A^4DE+95289831144601/3267280800000*A^3BDE+1076710706169299/34850995200000*A^2B^2DE+1123811346670067/52276492800000*AB^3DE+62780406647/6220800000*B^4DE+638884672826689/26138246400000*A^3CDE+144524458159147/4356374400000*A^2BCDE+241438955130817/8712748800000*AB^2CDE+45924188443/3110400000*B^3CDE+1234805514529073/52276492800000*A^2C^2DE+23580038122639/968083200000*ABC^2DE+95568973913/6220800000*B^2C^2DE+804752922640337/52276492800000*AC^3DE+35668631299/3110400000*BC^3DE+384481013/57600000*C^4DE+74098944545497/3920736960000*A^3D^2E+2654638675427621/104552985600000*A^2BD^2E+276319882150547/13069123200000*AB^2D^2E+424356929717/37324800000*B^3D^2E+3324322623479471/156829478400000*A^2CD^2E+23971036878001/1089093600000*ABCD^2E+257947333427/18662400000*B^2CD^2E+1251204977862227/78414739200000*AC^2D^2E+55507040701/4665600000*BC^2D^2E+19406574299/2488320000*C^3D^2E+61996677958057/4182119424000*A^2D^3E+53198078603431/3485099520000*ABD^3E+11964846313/1244160000*B^2D^3E+661173887580457/52276492800000*ACD^3E+2938752887/311040000*BCD^3E+7245724871/1036800000*C^2D^3E+115236794714537/13940398080000*AD^4E+5690090581/921600000*BD^4E+1535679677/307200000*CD^4E+4193357/1327104*D^5E+338271059367947/23233996800000*A^4E^2+1650773582550629/69701990400000*A^3BE^2+1732816268734339/69701990400000*A^2B^2E^2+1210499093468333/69701990400000*AB^3E^2+34424336009/4147200000*B^4E^2+6224461247183723/313658956800000*A^3CE^2+833568160397701/31365895680000*A^2BCE^2+1735236175783253/78414739200000*AB^2CE^2+445360693169/37324800000*B^3CE^2+220472095058371/11616998400000*A^2C^2E^2+1014765575815889/52276492800000*ABC^2E^2+50997682799/4147200000*B^2C^2E^2+778172118052271/62731791360000*AC^3E^2+2755074413/298598400*BC^3E^2+34039795237/6220800000*C^4E^2+5381548714495441/313658956800000*A^3DE^2+1214011805665579/52276492800000*A^2BDE^2+1012125259250681/52276492800000*AB^2DE^2+386059918111/37324800000*B^3DE^2+152110032339827/7841473920000*A^2CDE^2+17635340794933/871274880000*ABCDE^2+14780359741/1166400000*B^2CDE^2+2293669885690931/156829478400000*AC^2DE^2+101880005143/9331200000*BC^2DE^2+17681194279/2488320000*C^3DE^2+105772842257933/6970199040000*A^2D^2E^2+18277703634263/1161699840000*ABD^2E^2+8192854481/829440000*B^2D^2E^2+56800134283339/4356374400000*ACD^2E^2+674099251/69120000*BCD^2E^2+9942252677/1382400000*C^2D^2E^2+118019254686551/12546358272000*AD^3E^2+262291085231/37324800000*BD^3E^2+70759883047/12441600000*CD^3E^2+866915/221184*D^4E^2+304974813919019/23233996800000*A^3E^3+2720862539173223/156829478400000*A^2BE^3+9019498016606407/627317913600000*AB^2E^3+97615952077/12441600000*B^3E^3+2272732173118313/156829478400000*A^2CE^3+38998100283913/2613824640000*ABCE^3+87758437249/9331200000*B^2CE^3+679381693063673/62731791360000*AC^2E^3+301060083247/37324800000*BC^2E^3+3334123807/622080000*C^3E^3+980512282755091/78414739200000*A^2DE^3+113200734700439/8712748800000*ABDE^3+15190219751/1866240000*B^2DE^3+140820826766359/13069123200000*ACDE^3+2508838199/311040000*BCDE^3+18439632743/3110400000*C^2DE^3+179477665272203/20910597120000*AD^2E^3+79848807871/12441600000*BD^2E^3+21551881519/4147200000*CD^2E^3+873649/221184*D^3E^3+1247987598607873/139403980800000*A^2E^4+1278475172085437/139403980800000*ABE^4+47982131939/8294400000*B^2E^4+1590627767373679/209105971200000*ACE^4+15707074159/2764800000*BCE^4+13051340477/3110400000*C^2E^4+1366674481993589/209105971200000*ADE^4+121791148661/24883200000*BDE^4+3657896221/921600000*CDE^4+1436645/442368*D^2E^4+3148925795230919/697019904000000*AE^5+420429927127/124416000000*BE^5+21318513797/7776000000*CE^5+3085789/1327104*DE^5+24955/15552*E^6+2391882175671533/209105971200000*A^5F+21214856188103/968083200000*A^4BF+290594046665593/10455298560000*A^3B^2F+315924910853137/13069123200000*A^2B^3F+170789161553897/11616998400000*AB^4F+156839465237/24883200000*B^5F+641728006059697/34850995200000*A^4CF+49509233621653/1633640400000*A^3BCF+13790184050821/430259200000*A^2B^2CF+1166817497659309/52276492800000*AB^3CF+21847463939/2073600000*B^4CF+6697254886626911/313658956800000*A^3C^2F+65908703727053/2323399680000*A^2BC^2F+34251312399733/1452124800000*AB^2C^2F+239025748927/18662400000*B^3C^2F+1100825741535761/62731791360000*A^2C^3F+2491407846499/139403980800*ABC^3F+211653484469/18662400000*B^2C^3F+353788359429107/34850995200000*AC^4F+783575177/103680000*BC^4F+6722648737/1658880000*C^5F+1667990725085039/104552985600000*A^4DF+4162586888112581/156829478400000*A^3BDF+2943710468973793/104552985600000*A^2B^2DF+3070361051785801/156829478400000*AB^3DF+56904444611/6220800000*B^4DF+698135269753013/31365895680000*A^3CDF+264340458597067/8712748800000*A^2BCDF+147269975180939/5808499200000*AB^2CDF+1962348311/145800000*B^3CDF+1689085832223677/78414739200000*A^2C^2DF+388482863535031/17425497600000*ABC^2DF+261768849013/18662400000*B^2C^2DF+2199973082451337/156829478400000*AC^3DF+9761421257/933120000*BC^3DF+18832147663/3110400000*C^4DF+134484179630473/7841473920000*A^3D^2F+7260923050617169/313658956800000*A^2BD^2F+6049510020966893/313658956800000*AB^2D^2F+192764510771/18662400000*B^3D^2F+2021762274577417/104552985600000*A^2CD^2F+175573400257769/8712748800000*ABCD^2F+39265448341/3110400000*B^2CD^2F+4568601222293449/313658956800000*AC^2D^2F+50723178737/4665600000*BC^2D^2F+44114776433/6220800000*C^3D^2F+42198511992557/3136589568000*A^2D^3F+727113369195247/52276492800000*ABD^3F+20386706921/2332800000*B^2D^3F+1808483223783961/156829478400000*ACD^3F+20117274107/2332800000*BCD^3F+9886262267/1555200000*C^2D^3F+34802442952643/4646799360000*AD^4F+17419767463/3110400000*BD^4F+56462515207/12441600000*CD^4F+11333455/3981312*D^5F+246008960815333/17425497600000*A^4EF+742452208257719/31365895680000*A^3BEF+657744370080947/26138246400000*A^2B^2EF+685301672873987/39207369600000*AB^3EF+5043270491/622080000*B^4EF+3115743839848547/156829478400000*A^3CEF+237458197442129/8712748800000*A^2BCEF+132388994166193/5808499200000*AB^2CEF+56139856997/4665600000*B^3CEF+151110608539039/7841473920000*A^2C^2EF+349417749540089/17425497600000*ABC^2EF+58649827733/4665600000*B^2C^2EF+393178545995527/31365895680000*AC^3EF+87365941663/9331200000*BC^3EF+8354482339/1555200000*C^4EF+5992979161819/348509952000*A^3DEF+20767964058823/871274880000*A^2BDEF+34777715936917/1742549760000*AB^2DEF+10826809231/1036800000*B^3DEF+4343943497653/217818720000*A^2CDEF+51185279267/2420208000*ABCDEF+135954721/10368000*B^2CDEF+21966976119281/1452124800000*AC^2DEF+1959554611/172800000*BC^2DEF+2492056847/345600000*C^3DEF+242241143263921/15682947840000*A^2D^2EF+70954351490741/4356374400000*ABD^2EF+5903160557/583200000*B^2D^2EF+78537418314679/5808499200000*ACD^2EF+389976041/38400000*BCD^2EF+11513885831/1555200000*C^2D^2EF+149548276993373/15682947840000*AD^3EF+66753652609/9331200000*BD^3EF+18051953717/3110400000*CD^3EF+320411/82944*D^4EF+247618606729889/17425497600000*A^3E^2F+380429856477137/19603684800000*A^2BE^2F+5079018323481289/313658956800000*AB^2E^2F+7120169287/829440000*B^3E^2F+2545427072650523/156829478400000*A^2CE^2F+74305073521187/4356374400000*ABCE^2F+49578777227/4665600000*B^2CE^2F+1920943934572949/156829478400000*AC^2E^2F+171026592611/18662400000*BC^2E^2F+73487715301/12441600000*C^3E^2F+293044139125979/20910597120000*A^2DE^2F+51797647637243/3485099520000*ABDE^2F+28627109497/3110400000*B^2DE^2F+107570342003813/8712748800000*ACDE^2F+9627821273/1036800000*BCDE^2F+4659842341/691200000*C^2DE^2F+610815854254607/62731791360000*AD^2E^2F+136506062399/18662400000*BD^2E^2F+18465877703/3110400000*CD^2E^2F+322651/73728*D^3E^2F+13829534509451/1290777600000*A^2E^3F+585494041621313/52276492800000*ABE^3F+2716284949/388800000*B^2E^3F+182300641966699/19603684800000*ACE^3F+65074531459/9331200000*BCE^3F+10566350221/2073600000*C^2E^3F+125353005037147/15682947840000*ADE^3F+28045156531/4665600000*BDE^3F+3797629003/777600000*CDE^3F+109445/27648*D^2E^3F+11195633953751/1936166400000*AE^4F+107942826943/24883200000*BE^4F+87658734919/24883200000*CE^4F+3965605/1327104*DE^4F+65905/31104*E^5F+1290482243785601/104552985600000*A^4F^2+1059834312198223/52276492800000*A^3BF^2+24798539851123/1161699840000*A^2B^2F^2+778400286158939/52276492800000*AB^3F^2+87721254761/12441600000*B^4F^2+2666683583290313/156829478400000*A^3CF^2+23998073482303/1045529856000*A^2BCF^2+1000150112467543/52276492800000*AB^2CF^2+95595382981/9331200000*B^3CF^2+11371665426799/697019904000*A^2C^2F^2+36582194271661/2178187200000*ABC^2F^2+2196728359/207360000*B^2C^2F^2+208735045584599/19603684800000*AC^3F^2+37017470917/4665600000*BC^3F^2+19303159001/4147200000*C^4F^2+4614602200316813/313658956800000*A^3DF^2+629483163489209/31365895680000*A^2BDF^2+5253292294674961/313658956800000*AB^2DF^2+10366027651/1166400000*B^3DF^2+5262668783388607/313658956800000*A^2CDF^2+3835558214161/217818720000*ABCDF^2+204954231251/18662400000*B^2CDF^2+1324258164368899/104552985600000*AC^2DF^2+29459574193/3110400000*BC^2DF^2+38036558371/6220800000*C^3DF^2+45506335415879/3485099520000*A^2D^2F^2+1424326187482867/104552985600000*ABD^2F^2+1177441573/138240000*B^2D^2F^2+141767723041577/12546358272000*ACD^2F^2+78997341731/9331200000*BCD^2F^2+515180057/82944000*C^2D^2F^2+506989709624921/62731791360000*AD^3F^2+56438561221/9331200000*BD^3F^2+30488540593/6220800000*CD^3F^2+6643987/1990656*D^4F^2+226625225224303/17425497600000*A^3EF^2+2803346453613473/156829478400000*A^2BEF^2+4683138762254531/313658956800000*AB^2EF^2+241800797/30720000*B^3EF^2+117288636914101/7841473920000*A^2CEF^2+137537929311443/8712748800000*ABCEF^2+45763427087/4665600000*B^2CEF^2+35454166607761/3136589568000*AC^2EF^2+158002955831/18662400000*BC^2EF^2+67469686277/12441600000*C^3EF^2+810471924147923/62731791360000*A^2DEF^2+26640337842079/1936166400000*ABDEF^2+3172029553/373248000*B^2DEF^2+4428226227161/387233280000*ACDEF^2+1487820509/172800000*BCDEF^2+310137107/49766400*C^2DEF^2+188015940715909/20910597120000*AD^2EF^2+42064086211/6220800000*BD^2EF^2+2847056177/518400000*CD^2EF^2+889553/221184*D^3EF^2+25069406019811/2323399680000*A^2E^2F^2+1188664098372209/104552985600000*ABE^2F^2+5857845637/829440000*B^2E^2F^2+370228428540689/39207369600000*ACE^2F^2+132369778289/18662400000*BCE^2F^2+856426739/165888000*C^2E^2F^2+814704082861/100370866176*ADE^2F^2+114108354577/18662400000*BDE^2F^2+15462288247/3110400000*CDE^2F^2+98707/24576*D^2E^2F^2+14846198071/2323399680*AE^3F^2+59689146769/12441600000*BE^3F^2+16156943467/4147200000*CE^3F^2+730681/221184*DE^3F^2+78575/31104*E^4F^2+1125092632047677/104552985600000*A^3F^3+62620611491813/4356374400000*A^2BF^3+25985817616333/2178187200000*AB^2F^3+80245589837/12441600000*B^3F^3+314170497114071/26138246400000*A^2CF^3+18106773281029/1452124800000*ABCF^3+6081514229/777600000*B^2CF^3+470439725157853/52276492800000*AC^2F^3+20890536149/3110400000*BC^2F^3+18308331797/4147200000*C^3F^3+1085228621413669/104552985600000*A^2DF^3+10518873009523/968083200000*ABDF^3+42140458997/6220800000*B^2DF^3+58945519849747/6534561600000*ACDF^3+1169129843/172800000*BCDF^3+15374840971/3110400000*C^2DF^3+49769901058723/6970199040000*AD^2F^3+33282112087/6220800000*BD^2F^3+6746714471/1555200000*CD^2F^3+6486361/1990656*D^3F^3+319355828611439/34850995200000*A^2EF^3+33648407874997/3485099520000*ABEF^3+1553487103/259200000*B^2EF^3+209710161595399/26138246400000*ACEF^3+18749358887/3110400000*BCEF^3+27248746777/6220800000*C^2EF^3+72142526345663/10455298560000*ADEF^3+3234040789/622080000*BDEF^3+548018821/129600000*CDEF^3+282653/82944*D^2EF^3+75694172833/12907776000*AE^2F^3+54827336137/12441600000*BE^2F^3+44543895473/12441600000*CE^2F^3+2015027/663552*DE^2F^3+78575/31104*E^3F^3+1500886766013893/209105971200000*A^2F^4+258526885021/34850995200*ABF^4+115760478533/24883200000*B^2F^4+322023226980481/52276492800000*ACF^4+28682789737/6220800000*BCF^4+28048796573/8294400000*C^2F^4+1107743916133987/209105971200000*ADF^4+16488894401/4147200000*BDF^4+4017301183/1244160000*CDF^4+10437469/3981312*D^2F^4+18061240044649/3872332800000*AEF^4+87274654999/24883200000*BEF^4+14194591607/4976640000*CEF^4+357077/147456*DEF^4+65905/31104*E^2F^4+3713434169657939/1045529856000000*AF^5+331328208659/124416000000*BF^5+89747579579/41472000000*CF^5+7316015/3981312*DF^5+24955/15552*EF^5+315/256*F^6
        +28960904587501/344207360000*A^5+155326218791899/929359872000*A^4B+11053322829041/51631104000*A^3B^2+19250787327013/103262208000*A^2B^3+209177251484233/1858719744000*AB^4+125282339631/2689120000*B^5+1969172086724093/13940398080000*A^4C+1679504530982771/6970199040000*A^3BC+1193926267748783/4646799360000*A^2B^2C+310434796639249/1742549760000*AB^3C+14174196687563/174254976000*B^4C+1153262382122971/6970199040000*A^3C^2+635596167870521/2788079616000*A^2BC^2+106067832473273/557615923200*AB^2C^2+87199585232233/871274880000*B^3C^2+1895446456726843/13940398080000*A^2C^3+13855233984197/96808320000*ABC^3+77358567866777/871274880000*B^2C^3+724602684215677/9293598720000*AC^4+3405637674161/58084992000*BC^4+649438637/21600000*C^5+1714531058721541/13940398080000*A^4D+20095069502297/94832640000*A^3BD+50142941322763/221276160000*A^2B^2D+104155287913123/663828480000*AB^3D+123665289140983/1742549760000*B^4D+2491376831783477/13940398080000*A^3CD+97572233440471/387233280000*A^2BCD+36353576180893/172103680000*AB^2CD+189823508331701/1742549760000*B^3CD+810298224650479/4646799360000*A^2C^2D+96144910768393/516311040000*ABC^2D+66612867467743/580849920000*B^2C^2D+1574512995192709/13940398080000*AC^3D+148225699018987/1742549760000*BC^3D+407850521/8640000*C^4D+931175241597611/6970199040000*A^3D^2+24803999459521/132765696000*A^2BD^2+4147404174931/26553139200*AB^2D^2+141377829239377/1742549760000*B^3D^2+2188086535457243/13940398080000*A^2CD^2+13064555791037/77446656000*ABCD^2+22540366782877/217818720000*B^2CD^2+1654297992261709/13940398080000*AC^2D^2+155881162718011/1742549760000*BC^2D^2+121104161/2160000*C^3D^2+1459206046300643/13940398080000*A^2D^3+4945567699097/44255232000*ABD^3+119707003365413/1742549760000*B^2D^3+16039652600891/172103680000*ACD^3+302821778201/4302592000*BCD^3+870413867/17280000*C^2D^3+59579060716901/1032622080000*AD^4+8436742364017/193616640000*BD^4+409786913/11520000*CD^4+203367533/9600000*D^5+94055577057987/860518400000*A^4E+13220035717622627/69701990400000*A^3BE+4722336641991469/23233996800000*A^2B^2E+9799206748064479/69701990400000*AB^3E+183440131613971/2904249600000*B^4E+6695310769126447/41821194240000*A^3CE+3956492201799133/17425497600000*A^2BCE+13276302259833101/69701990400000*AB^2CE+2554621982879777/26138246400000*B^3CE+2182377191307427/13940398080000*A^2C^2E+3903371320798357/23233996800000*ABC^2E+898560853984859/8712748800000*B^2C^2E+4236168112518239/41821194240000*AC^3E+1996941863621959/26138246400000*BC^3E+1816954753/43200000*C^4E+2911496335145119/20910597120000*A^3DE+6948466713519517/34850995200000*A^2BDE+2917888254799603/17425497600000*AB^2DE+556878239476291/6534561600000*B^3DE+1948651471013131/11616998400000*A^2CDE+14179383867641/77446656000*ABCDE+40415302384099/363031200000*B^2CDE+2224430285306177/17425497600000*AC^2DE+105210623169871/1089093600000*BC^2DE+7673774573/129600000*C^3DE+1758086320266587/13940398080000*A^2D^2E+3183797047300519/23233996800000*ABD^2E+727103034927103/8712748800000*B^2D^2E+7974817481503693/69701990400000*ACD^2E+755796930565483/8712748800000*BCD^2E+5314021327/86400000*C^2D^2E+808941815941447/10455298560000*AD^3E+191482909374241/3267280800000*BD^3E+6206565301/129600000*CD^3E+145758401/4800000*D^4E+322958826236597/2904249600000*A^3E^2+10958709966149657/69701990400000*A^2BE^2+9175350384098401/69701990400000*AB^2E^2+196726338513829/2904249600000*B^3E^2+5530696577278417/41821194240000*A^2CE^2+7704449348651/53782400000*ABCE^2+571350754812983/6534561600000*B^2CE^2+837862963753321/8364238848000*AC^2E^2+1978268273969851/26138246400000*BC^2E^2+1013296501/21600000*C^3E^2+4798461547294403/41821194240000*A^2DE^2+2912026868386999/23233996800000*ABDE^2+1988707169174599/26138246400000*B^2DE^2+2432898981528431/23233996800000*ACDE^2+230878834841371/2904249600000*BCDE^2+14554998511/259200000*C^2DE^2+3347701486891189/41821194240000*AD^2E^2+793484171815921/13069123200000*BD^2E^2+12865626053/259200000*CD^2E^2+754775843/21600000*D^3E^2+1943922806612879/23233996800000*A^2E^3+233007446066797/2581555200000*ABE^3+160074125661661/2904249600000*B^2E^3+1050860491954283/13940398080000*ACE^3+497346927499663/8712748800000*BCE^3+129670573/3200000*C^2E^3+28365496869593/435637440000*ADE^3+53841086352637/1089093600000*BDE^3+109226677/2700000*CDE^3+684408257/21600000*D^2E^3+2075513107262837/46467993600000*AE^4+98278110577159/2904249600000*BE^4+4783616003/172800000*CE^4+1020366851/43200000*DE^4+145685/9216*E^5+10227479358119/103723200000*A^4F+11990298468481217/69701990400000*A^3BF+2144870025948017/11616998400000*A^2B^2F+8894821412558929/69701990400000*AB^3F+20709025007371/363031200000*B^4F+30382021792662403/209105971200000*A^3CF+1803545590381571/8712748800000*A^2BCF+12110361921897299/69701990400000*AB^2CF+1160633013385373/13069123200000*B^3CF+4959782685184723/34850995200000*A^2C^2F+3562009115203573/23233996800000*ABC^2F+51120394621577/544546800000*B^2C^2F+19239307162563251/209105971200000*AC^3F+908003443995481/13069123200000*BC^3F+821166833/21600000*C^4F+1057493670254603/8364238848000*A^3DF+2376532194612673/13069123200000*A^2BDF+31952359642018199/209105971200000*AB^2DF+2025064163471201/26138246400000*B^3DF+2286427456941043/14936140800000*A^2CDF+77906573925217/464679936000*ABCDF+1329380767785181/13069123200000*B^2CDF+3482272515850403/29872281600000*AC^2DF+2308308576122051/26138246400000*BC^2DF+931003783/17280000*C^3DF+799595790027743/6970199040000*A^2D^2F+2906801720860691/23233996800000*ABD^2F+165562354565003/2178187200000*B^2D^2F+446044382415899/4267468800000*ACD^2F+1036777676321803/13069123200000*BCD^2F+2906807539/51840000*C^2D^2F+2940872346701447/41821194240000*AD^3F+1393852851722977/26138246400000*BD^3F+2260769609/51840000*CD^3F+3559590377/129600000*D^4F+57978119514191/516311040000*A^3EF+3098007095689/19051200000*A^2BEF+583620433028023/4267468800000*AB^2EF+200219342126527/2904249600000*B^3EF+14324813299829873/104552985600000*A^2CEF+350415023844061/2323399680000*ABCEF+297993035587007/3267280800000*B^2CEF+21839438230853293/209105971200000*AC^2EF+2071338068293003/26138246400000*BC^2EF+12443629649/259200000*C^3EF+138247500549373/1161699840000*A^2DEF+102292260633271/774466560000*ABDEF+23075380314149/290424960000*B^2DEF+256851262538021/2323399680000*ACDEF+2447632182383/29042496000*BCDEF+424145377/7200000*C^2DEF+3499798084583681/41821194240000*AD^2EF+66644107029169/1045529856000*BD^2EF+13540395181/259200000*CD^2EF+387806243/10800000*D^3EF+221600244090661/2323399680000*A^2E^2F+348385472154829/3319142400000*ABE^2F+479548504103/7563150000*B^2E^2F+18360963839362243/209105971200000*ACE^2F+109138435341673/1633640400000*BCE^2F+12152737519/259200000*C^2E^2F+634957836141269/8364238848000*ADE^2F+302751816223441/5227649280000*BDE^2F+3078568589/64800000*CDE^2F+397525673/10800000*D^2E^2F+259888347783959/4646799360000*AE^3F+370692213061747/8712748800000*BE^3F+251045773/7200000*CE^3F+321536701/10800000*DE^3F+867485/41472*E^4F+314218762661777/3319142400000*A^3F^2+9408383212647527/69701990400000*A^2BF^2+985793900502017/8712748800000*AB^2F^2+20984010898009/363031200000*B^3F^2+23763850734342253/209105971200000*A^2CF^2+159883156691501/1290777600000*ABCF^2+246044653725107/3267280800000*B^2CF^2+281668447649951/3267280800000*AC^2F^2+426403313745947/6534561600000*BC^2F^2+288718759/7200000*C^3F^2+165065609158873/1672847769600*A^2DF^2+7558449205169473/69701990400000*ABDF^2+1714071692493971/26138246400000*B^2DF^2+902956405337551/9957427200000*ACDF^2+600837774624397/8712748800000*BCDF^2+16753313/345600*C^2DF^2+36050508757447/522764928000*AD^2F^2+1369619106618073/26138246400000*BD^2F^2+1111712113/25920000*CD^2F^2+215345131/7200000*D^3F^2+406583855484587/4646799360000*A^2EF^2+963175854683687/9957427200000*ABEF^2+169241611507727/2904249600000*B^2EF^2+1881135963762227/23233996800000*ACEF^2+179112866064527/2904249600000*BCEF^2+5591426797/129600000*C^2EF^2+97619903068607/1394039808000*ADEF^2+23299691636933/435637440000*BDEF^2+3793544677/86400000*CDEF^2+244087807/7200000*D^2EF^2+5299306130827/92935987200*AE^2F^2+14020940707339/322694400000*BE^2F^2+1026219973/28800000*CE^2F^2+328681639/10800000*DE^2F^2+161645/6912*E^3F^2+57050955314527/829785600000*A^2F^3+579061666750711/7744665600000*ABF^3+1221593225571/26891200000*B^2F^3+484177430401783/7744665600000*ACF^3+7654792391521/161347200000*BCF^3+1445822111/43200000*C^2F^3+753462131507539/13940398080000*ADF^3+119433533929823/2904249600000*BDF^3+34940239/1036800*CDF^3+1698527371/64800000*D^2F^3+222062428179689/4646799360000*AEF^3+317395636249037/8712748800000*BEF^3+7747905743/259200000*CEF^3+183921559/7200000*DEF^3+885755/41472*E^2F^3+237211880355049/6638284800000*AF^4+39408995384011/1452124800000*BF^4+640445531/28800000*CF^4+2461937003/129600000*DF^4+1382215/82944*EF^4+10615/864*F^5+52265182780871/580849920000*A^4G+135647992891721/860518400000*A^3BG+72883666162623/430259200000*A^2B^2G+302073370703771/2581555200000*AB^3G+9457276443101/181515600000*B^4G+3095052102938483/23233996800000*A^3CG+110635316014367/580849920000*A^2BCG+743204191164773/4646799360000*AB^2CG+118348289252359/1452124800000*B^3CG+1517720280225829/11616998400000*A^2C^2G+218668173124243/1548933120000*ABC^2G+1957466170441/22689450000*B^2C^2G+217913106883099/2581555200000*AC^3G+30882381043261/484041600000*BC^3G+125083817/3600000*C^4G+1616603618983813/13940398080000*A^3DG+62499819665503/373403520000*A^2BDG+840658476064949/5974456320000*AB^2DG+123947061527491/1742549760000*B^3DG+2947797882905473/20910597120000*A^2CDG+71942728066457/464679936000*ABCDG+245083570967029/2613824640000*B^2CDG+4491856675056089/41821194240000*AC^2DG+425731623694283/5227649280000*BC^2DG+12830390009/259200000*C^3DG+734408274119771/6970199040000*A^2D^2G+8500847225681/73758720000*ABD^2G+30444792745351/435637440000*B^2D^2G+4028808548088551/41821194240000*ACD^2G+191276260453483/2613824640000*BCD^2G+13373470549/259200000*C^2D^2G+899781191571301/13940398080000*AD^3G+9485650410559/193616640000*BD^3G+1154650277/28800000*CD^3G+3254964967/129600000*D^4G+478780950614591/4646799360000*A^3EG+195674475148591/1306912320000*A^2BEG+5268216864819331/41821194240000*AB^2EG+12258950165017/193616640000*B^3EG+2639086681884253/20910597120000*A^2CEG+64733443264753/464679936000*ABCEG+27476386393229/326728080000*B^2CEG+1341855803486459/13940398080000*AC^2EG+127376153414341/1742549760000*BC^2EG+11436460693/259200000*C^3EG+229315447285669/2091059712000*A^2DEG+10501577921607/86051840000*ABDEG+191561261168281/2613824640000*B^2DEG+237431000668157/2323399680000*ACDEG+11321813632459/145212480000*BCDEG+176177609/3240000*C^2DEG+3227058094918457/41821194240000*AD^2EG+307511125744589/5227649280000*BD^2EG+2500633169/51840000*CD^2EG+356617853/10800000*D^3EG+22624604638129/258155520000*A^2E^2G+149822988691709/1548933120000*ABE^2G+941007826271/16134720000*B^2E^2G+3385712396915071/41821194240000*ACE^2G+8056758627559/130691232000*BCE^2G+11186867003/259200000*C^2E^2G+2928272429186561/41821194240000*ADE^2G+279475971628697/5227649280000*BDE^2G+1137427387/25920000*CDE^2G+732306241/21600000*D^2E^2G+238640091213919/4646799360000*AE^3G+68139181037671/1742549760000*BE^3G+51306603/1600000*CE^3G+147926773/5400000*DE^3G+793585/41472*E^4G+4406179561547/47416320000*A^3FG+4730035832718661/34850995200000*A^2BFG+1991060357756749/17425497600000*AB^2FG+41518991281907/726062400000*B^3FG+11969468587564547/104552985600000*A^2CFG+49123097785177/387233280000*ABCFG+499166147845711/6534561600000*B^2CFG+4567984121515373/52276492800000*AC^2FG+434142913541783/6534561600000*BC^2FG+574425989/14400000*C^3FG+2081225000260901/20910597120000*A^2DFG+86104323708847/774466560000*ABDFG+174095900620667/2613824640000*B^2DFG+30920419064051/331914240000*ACDFG+4132965715759/58084992000*BCDFG+6411050659/129600000*C^2DFG+366437074790491/5227649280000*AD^2FG+139838918912527/2613824640000*BD^2FG+5690475317/129600000*CD^2FG+10083829/337500*D^3FG+68415986025841/774466560000*A^2EFG+10980998434409/110638080000*ABEFG+5735196511817/96808320000*B^2EFG+193404310240741/2323399680000*ACEFG+18495506866741/290424960000*BCEFG+634567387/14400000*C^2EFG+83753227434427/1161699840000*ADEFG+4015405269509/72606240000*BDEFG+6554869/144000*CDEFG+250171619/7200000*D^2EFG+67481513252567/1161699840000*AE^2FG+1434720985541/32269440000*BE^2FG+87741821/2400000*CE^2FG+337817713/10800000*DE^2FG+54005/2304*E^3FG+12375633232859/165957120000*A^2F^2G+213941047815023/2581555200000*ABF^2G+2257763065687/45378900000*B^2F^2G+1613251721817223/23233996800000*ACF^2G+76926608114879/1452124800000*BCF^2G+265581737/7200000*C^2F^2G+279288666405643/4646799360000*ADF^2G+26705862265567/580849920000*BDF^2G+3264883213/86400000*CDF^2G+313379683/10800000*D^2F^2G+247126025859283/4646799360000*AEF^2G+1578375267317/38723328000*BEF^2G+1449125333/43200000*CEF^2G+1240571027/43200000*DEF^2G+12145/512*E^2F^2G+13961566124621/331914240000*AF^3G+15540619847311/484041600000*BF^3G+2278492553/86400000*CF^3G+974402359/43200000*DF^3G+68425/3456*EF^3G+13075/864*F^4G+42338664390683/516311040000*A^3G^2+101291020143511/860518400000*A^2BG^2+15934315002359/161347200000*AB^2G^2+505861342943/10084200000*B^3G^2+256041500688317/2581555200000*A^2CG^2+5609360074589/51631104000*ABCG^2+1770563320611/26891200000*B^2CG^2+72921892800041/968083200000*AC^2G^2+3454914416263/60505200000*BC^2G^2+20909423/600000*C^3G^2+1201221836092843/13940398080000*A^2DG^2+189511704085339/1991485440000*ABDG^2+99970640150401/1742549760000*B^2DG^2+44407878988909/557615923200*ACDG^2+105680793694069/1742549760000*BCDG^2+2751716561/64800000*C^2DG^2+105087995991761/1742549760000*AD^2G^2+15993293249243/348509952000*BD^2G^2+101521057/2700000*CD^2G^2+562050013/21600000*D^3G^2+355238271593627/4646799360000*A^2EG^2+1183833509011469/13940398080000*ABEG^2+5925391075771/116169984000*B^2EG^2+36727875132527/516311040000*ACEG^2+31516841702299/580849920000*BCEG^2+4900435949/129600000*C^2EG^2+429099983387171/6970199040000*ADEG^2+20511080786131/435637440000*BDEG^2+668528167/17280000*CDEG^2+321207443/10800000*D^2EG^2+115939661785847/2323399680000*AE^2G^2+491516202959/12907776000*BE^2G^2+900266831/28800000*CE^2G^2+577109623/21600000*DE^2G^2+140755/6912*E^3G^2+326550716099/4741632000*A^2FG^2+1785300454514207/23233996800000*ABFG^2+8353924826063/181515600000*B^2FG^2+4489689707931709/69701990400000*ACFG^2+214308714018197/4356374400000*BCFG^2+245937407/7200000*C^2FG^2+777548221930729/13940398080000*ADFG^2+74426150138773/1742549760000*BDFG^2+1011634711/28800000*CDFG^2+290433593/10800000*D^2FG^2+76466723130041/1548933120000*AEFG^2+7333265655929/193616640000*BEFG^2+1347413323/43200000*CEFG^2+384670589/14400000*DEFG^2+101365/4608*E^2FG^2+1039815821637/24586240000*AF^2G^2+5218579174343/161347200000*BF^2G^2+255267383/9600000*CF^2G^2+327633007/14400000*DF^2G^2+92045/4608*EF^2G^2+4745/288*F^3G^2+2483301664661/43025920000*A^2G^3+489236186636063/7744665600000*ABG^3+192740486411/5042100000*B^2G^3+1228362208155421/23233996800000*ACG^3+58363233140153/1452124800000*BCG^3+22568631/800000*C^2G^3+127529312039743/2788079616000*ADG^3+20250112125911/580849920000*BDG^3+3706939699/129600000*CDG^3+716791603/32400000*D^2G^3+62680209127243/1548933120000*AEG^3+10769169953149/348509952000*BEG^3+6579740821/259200000*CEG^3+468977297/21600000*DEG^3+748375/41472*E^2G^3+4600199701/126443520*AFG^3+40398648016633/1452124800000*BFG^3+1976564293/86400000*CFG^3+845920219/43200000*DFG^3+59435/3456*EFG^3+13075/864*F^2G^3+90816677205313/3097866240000*AG^4+674876083349/30252600000*BG^4+87855329/4800000*CG^4+126768823/8100000*DG^4+1139615/82944*EG^4+10615/864*FG^4+315/32*G^5
        +17301822978871/22759833600*A^4+72864395936281/53106278400*A^3B+78869491607779/53106278400*A^2B^2+162838422339577/159318835200*AB^3+870881141979/1966899200*B^4+1105187656613/948326400*A^3C+22777139724131/13276569600*A^2BC+38362242663551/26553139200*AB^2C+282560696431/393379840*B^3C+10920330377/9483264*A^2C^2+33941268892799/26553139200*ABC^2+4527680420171/5900697600*B^2C^2+263368461583/355622400*AC^3+3332027154227/5900697600*BC^3+9116605013/30732800*C^4+30061805659339/29503488000*A^3D+66963959889781/44255232000*A^2BD+112908240153131/88510464000*AB^2D+247659693887/393379840*B^3D+28342436601859/22127616000*A^2CD+10641821579873/7375872000*ABCD+4225402774491/4917248000*B^2CD+1805210882891/1843968000*AC^2D+22088290139729/29503488000*BC^2D+6771468141/15366400*C^3D+247751980635409/265531392000*A^2D^2+278181245629849/265531392000*ABD^2+18422374393751/29503488000*B^2D^2+14608802236489/16595712000*ACD^2+19896653277601/29503488000*BCD^2+142670768061/307328000*C^2D^2+452867349106667/796594176000*AD^3+12841508413967/29503488000*BD^3+27544698113/76832000*CD^3+27550987/128000*D^4+603224189529733/663828480000*A^3E+64387673757103/47416320000*A^2BE+54325354965359/47416320000*AB^2E+27658427944691/49172480000*B^3E+76366984064897/66382848000*A^2CE+864231865439441/663828480000*ABCE+9503968723583/12293120000*B^2CE+233708688187417/265531392000*AC^2E+33145894631881/49172480000*BC^2E+1211576322819/3073280000*C^3E+133279531576967/132765696000*A^2DE+189966923996083/165957120000*ABDE+4158453774583/6146560000*B^2DE+639532341235243/663828480000*ACDE+18218867823187/24586240000*BCDE+3105925692479/6146560000*C^2DE+15685700656259/22127616000*AD^2E+26794385246277/49172480000*BD^2E+690972588503/1536640000*CD^2E+379449177/1280000*D^3E+172441514992631/221276160000*A^2E^2+581105856427/658560000*ABE^2+25727523239521/49172480000*B^2E^2+7300880434699/9834496000*ACE^2+84094583101643/147517440000*BCE^2+18721940001/48020000*C^2E^2+8561990021621/13276569600*ADE^2+73237260556531/147517440000*BDE^2+2520514000127/6146560000*CDE^2+394271667/1280000*D^2E^2+903967521403193/1991485440000*AE^3+17140387917657/49172480000*BE^3+884160153453/3073280000*CE^3+316213549/1280000*DE^3+42117903/256000*E^4+22278977086151/27095040000*A^3F+117072981141617/94832640000*A^2BF+197672479735427/189665280000*AB^2F+102290325763/200704000*B^3F+694708810403249/663828480000*A^2CF+197253570210671/165957120000*ABCF+25971830489/36879360*B^2CF+531914038319047/663828480000*AC^2F+18126008428447/29503488000*BC^2F+137363915033/384160000*C^3F+15163236401009/16595712000*A^2DF+173505529312337/165957120000*ABDF+464061788093/752640000*B^2DF+584469718513307/663828480000*ACDF+50002941840769/73758720000*BCDF+8500036641701/18439680000*C^2DF+171473241705553/265531392000*AD^2F+10472779632577/21073920000*BD^2F+1261321227343/3073280000*CD^2F+1033317647/3840000*D^3F+27015628762819/33191424000*A^2EF+88795138230977/94832640000*ABEF+40595442225547/73758720000*B^2EF+37422729947149/47416320000*ACEF+2137718968677/3512320000*BCEF+759863906761/1843968000*C^2EF+5696141671459/8297856000*ADEF+1565783462129/2950348800*BDEF+1350540639837/3073280000*CDEF+104539973/320000*D^2EF+71343140019257/132765696000*AE^2F+61167241838317/147517440000*BE^2F+105432850673/307328000*CE^2F+11799523/40000*DE^2F+162790037/768000*E^3F+42078360955859/63221760000*A^2F^2+15984113581333/21073920000*ABF^2+38532526817/86016000*B^2F^2+23557561949129/36879360000*ACF^2+2899682152103/5900697600*BCF^2+1027729557199/3073280000*C^2F^2+147460136615411/265531392000*ADF^2+429803510597/1003520000*BDF^2+6531337631873/18439680000*CDF^2+1015926407/3840000*D^2F^2+16371189165599/33191424000*AEF^2+56216473154897/147517440000*BEF^2+145464914323/460992000*CEF^2+173746233/640000*DEF^2+166911947/768000*E^2F^2+212208423841657/568995840000*AF^3+172851160379/602112000*BF^3+364652408559/1536640000*CF^3+783398089/3840000*DF^3+138120619/768000*EF^3+267215/2048*F^4+16647252535961/22127616000*A^3G+125428557246479/110638080000*A^2BG+70626321017783/73758720000*AB^2G+22954348720669/49172480000*B^3G+212762434158527/221276160000*A^2CG+48466746644669/44255232000*ABCG+6825099201091/10536960000*B^2CG+326001647215357/442552320000*AC^2G+11913541451881/21073920000*BC^2G+20146626733/61465600*C^3G+27874976068871/33191424000*A^2DG+127938525262907/132765696000*ABDG+557939970683/983449600*B^2DG+53898309841993/66382848000*ACDG+10548665507/16859136*BCDG+7827159184057/18439680000*C^2DG+9857686666753/16595712000*AD^2G+13498246292563/29503488000*BD^2G+2323903154577/6146560000*CD^2G+473901599/1920000*D^3G+99361350366559/132765696000*A^2EG+14326874434499/16595712000*ABEG+7473061285397/14751744000*B^2EG+1208204794679/1659571200*ACEG+1036118302993/1843968000*BCEG+3499691591737/9219840000*C^2EG+14017114958717/22127616000*ADEG+903807826223/1843968000*BDEG+62401616861/153664000*CDEG+72265543/240000*D^2EG+131303438975213/265531392000*AE^2G+11267424129851/29503488000*BE^2G+1943355409979/6146560000*CE^2G+261113711/960000*DE^2G+74698031/384000*E^3G+1283561689307/1896652800*A^2FG+65035326640457/82978560000*ABFG+33839969604403/73758720000*B^2FG+6860388637321/10372320000*ACFG+1570735984647/3073280000*BCFG+2115366194481/6146560000*C^2FG+5096681377367/8851046400*ADFG+411264555911/921984000*BDFG+2273463637411/6146560000*CDFG+52462549/192000*D^2FG+67968592599853/132765696000*AEFG+1953551715251/4917248000*BEFG+2027359255027/6146560000*CEFG+11370777/40000*DEFG+43214411/192000*E^2FG+1598179301011/3793305600*AF^2G+48099780510443/147517440000*BF^2G+415386146267/1536640000*CF^2G+44696597/192000*DF^2G+78888041/384000*EF^2G+324255/2048*F^3G+38385069795913/66382848000*A^2G^2+439684612725097/663828480000*ABG^2+57528030601187/147517440000*B^2G^2+741198054805841/1327656960000*ACG^2+9064464822191/21073920000*BCG^2+89596828457/307328000*C^2G^2+4030036174277/8297856000*ADG^2+11067245112109/29503488000*BDG^2+2863070969513/9219840000*CDG^2+110833151/480000*D^2G^2+114600190676539/265531392000*AEG^2+1970464701649/5900697600*BEG^2+5103824854637/18439680000*CEG^2+152516759/640000*DEG^2+72903161/384000*E^2G^2+2462213143307/6322176000*AFG^2+44512578465233/147517440000*BFG^2+1538709745633/6146560000*CFG^2+3451039/16000*DFG^2+24372697/128000*EFG^2+328245/2048*F^2G^2+62372745788333/199148544000*AG^3+35630361550709/147517440000*BG^3+61438320693/307328000*CG^3+165142109/960000*DG^3+58274797/384000*EG^3+279005/2048*FG^3+13615/128*G^4+184390813140929/265531392000*A^3H+27867464210233/26553139200*A^2BH+47092301561743/53106278400*AB^2H+4241125906281/9834496000*B^3H+14778419790937/16595712000*A^2CH+67486034134867/66382848000*ABCH+1771391717681/2950348800*B^2CH+22654638787807/33191424000*AC^2H+15465763231307/29503488000*BC^2H+232797026823/768320000*C^3H+11477630630231/14751744000*A^2DH+7919763721669/8851046400*ABDH+1552027477129/2950348800*B^2DH+16689176964907/22127616000*ACDH+2860017728427/4917248000*BCDH+96829929583/245862400*C^2DH+146204114163757/265531392000*AD^2H+12521650034587/29503488000*BD^2H+431379693631/1229312000*CD^2H+36525381/160000*D^3H+2877479479403/4148928000*A^2EH+5322543319307/6638284800*ABEH+6931215545273/14751744000*B^2EH+1122600837713/1659571200*ACEH+1926743434387/3687936000*BCEH+108266069239/307328000*C^2EH+78169685019833/132765696000*ADEH+336247085687/737587200*BDEH+11613403853/30732800*CDEH+107362451/384000*D^2EH+20293206116629/44255232000*AE^2H+10455976445507/29503488000*BE^2H+18043305373/61465600*CE^2H+194024327/768000*DE^2H+46077669/256000*E^3H+11897836944661/18966528000*A^2FH+483324574623347/663828480000*ABFH+31393800002491/73758720000*B^2FH+408040969398203/663828480000*ACFH+4382241157907/9219840000*BCFH+1963676711459/6146560000*C^2FH+17767907718187/33191424000*ADFH+191293344127/460992000*BDFH+6347856604561/18439680000*CDFH+162415687/640000*D^2FH+3950252412727/8297856000*AEFH+5453424934817/14751744000*BEFH+5662151211377/18439680000*CEFH+10589609/40000*DEFH+26767989/128000*E^2FH+4941631651651/12644352000*AF^2H+44649727377251/147517440000*BF^2H+1543151850937/6146560000*CF^2H+276853747/1280000*DF^2H+48879309/256000*EF^2H+300135/2048*F^3H+36273471169/63221760*A^2GH+24634850415481/36879360000*ABGH+28747773839899/73758720000*B^2GH+1981826318593/3512320000*ACGH+32212176187229/73758720000*BCGH+224968623839/768320000*C^2GH+16317485106149/33191424000*ADGH+5626894370873/14751744000*BDGH+729863432413/2304960000*CDGH+223440721/960000*D^2GH+58066046360971/132765696000*AEGH+5014670760949/14751744000*BEGH+1302494549047/4609920000*CEGH+38995189/160000*DEGH+18423323/96000*E^2GH+3745808559289/9483264000*AFGH+215950083089/702464000*BFGH+785996225319/3073280000*CFGH+35322857/160000*DFGH+3122473/16000*EFGH+166275/1024*F^2GH+642988363853/1896652800*AG^2H+12931720229333/49172480000*BG^2H+335580349469/1536640000*CG^2H+180769403/960000*DG^2H+15968269/96000*EG^2H+306075/2048*FG^2H+15645/128*G^3H+135217017928709/265531392000*A^2H^2+31107340032409/53106278400*ABH^2+3383116441101/9834496000*B^2H^2+8199280893341/16595712000*ACH^2+11244511226533/29503488000*BCH^2+197823963033/768320000*C^2H^2+2537549088839/5900697600*ADH^2+9811558992677/29503488000*BDH^2+169357606837/614656000*CDH^2+65316927/320000*D^2H^2+50759430562667/132765696000*AEH^2+582549426107/1966899200*BEH^2+30203061319/122931200*CEH^2+54187093/256000*DEH^2+42995979/256000*E^2H^2+13091834597509/37933056000*AFH^2+13164487929587/49172480000*BFH^2+683166971671/3073280000*CFH^2+735920021/3840000*DFH^2+129998447/768000*EFH^2+290565/2048*F^2H^2+66421541519/210739200*AGH^2+12034816103173/49172480000*BGH^2+312501772969/1536640000*CGH^2+168411223/960000*DGH^2+29762893/192000*EGH^2+285315/2048*FGH^2+15645/128*G^2H^2+213162455249369/796594176000*AH^3+2032631851947/9834496000*BH^3+131571562401/768320000*CH^3+47190469/320000*DH^3+33324963/256000*EH^3+239445/2048*FH^3+13615/128*GH^3+2835/32*H^4
        +27031118935/4064256*A^3+34881580883/3386880*A^2B+23638910741/2709504*AB^2+67599233579/16257024*B^3+39021983729/4445280*A^2C+485650508207/47416320*ABC+848520040301/142248960*B^2C+1919972883469/284497920*AC^2+742896374071/142248960*BC^2+4126515251/1404928*C^3+22351772999/2903040*A^2D+2573290017281/284497920*ABD+746232279533/142248960*B^2D+1089554291867/142248960*ACD+1692790266103/284497920*BCD+2081878405/526848*C^2D+88890369593/16257024*AD^2+1208012628941/284497920*BD^2+14881287539/4214784*CD^2+585548099/263424*D^3+1632586775839/237081600*A^2E+235992097871/29030400*ABE+3343494720157/711244800*B^2E+5185455733/752640*ACE+155599057601/29030400*BCE+46704550511/13171200*C^2E+2140923955771/355622400*ADE+6677912588261/1422489600*BDE+68713168887/17561600*CDE+6209121213/2195200*D^2E+4334953794419/948326400*AE^2+2531257465739/711244800*BE^2+312360137899/105369600*CE^2+5625969549/2195200*DE^2+6755843/3840*E^3+317361380197/50803200*A^2F+20889903427/2822400*ABF+43392947683/10160640*B^2F+318894930871/50803200*ACF+49672667713/10160640*BCF+252830131/78400*C^2F+7806301939871/1422489600*ADF+190436658007/44452800*BDF+3922070139/1097600*CDF+6781930649/2634240*D^2F+92872295107/18966528*AEF+5446667865383/1422489600*BEF+3509547341/1097600*CEF+12156278059/4390400*DEF+4090987/1920*E^2F+397221368401/101606400*AF^2+619740712567/203212800*BF^2+12763194607/5017600*CF^2+5798753879/2634240*DF^2+936289/480*EF^2+3679463/2560*F^3+81488869163/14224896*A^2G+2420651390077/355622400*ABG+2788673196313/711244800*B^2G+41918143517/7257600*ACG+400280541641/88905600*BCG+78057183289/26342400*C^2G+479059489507/94832640*ADG+140363681333/35562240*BDG+43389038723/13171200*CDG+10392847117/4390400*D^2G+6414084385243/1422489600*AEG+5019854080937/1422489600*BEG+38838089261/13171200*CEG+33647424299/13171200*DEG+313629/160*E^2G+414983339443/101606400*AFG+910522138633/284497920*BFG+28203270649/10536960*CFG+2546910083/1097600*DFG+3952717/1920*EFG+6392819/3840*F^2G+3024384955/889056*AG^2+236302755733/88905600*BG^2+233843821703/105369600*CG^2+25316318933/13171200*DG^2+3272209/1920*EG^2+5899187/3840*FG^2+1846775/1536*G^3+167488154723/31610880*A^2H+224401100699/35562240*ABH+258162492409/71124480*B^2H+15874358401/2963520*ACH+297200238469/71124480*BCH+144621601471/52684800*C^2H+333334929941/71124480*ADH+521259716507/142248960*BDH+5373774439/1756160*CDH+2408234343/1097600*D^2H+2976188316769/711244800*AEH+582722076739/177811200*BEH+721716019/263424*CEH+3127513573/1317120*DEH+2326611/1280*E^2H+28533802933/7526400*AFH+1057217146343/355622400*BFH+15920013/6400*CFH+28414577813/13171200*DFH+1838009/960*EFH+5930269/3840*F^2H+70578075637/20321280*AGH+3877886107961/1422489600*BGH+60129787691/26342400*CGH+1739205901/878080*DGH+1125493/640*EGH+1015529/640*FGH+85575/64*G^2H+189711473743/63221760*AH^2+667881301583/284497920*BH^2+206723982301/105369600*CH^2+14580168/8575*DH^2+11584633/7680*EH^2+5223487/3840*FH^2+635775/512*GH^2+49175/48*H^3+40111361755/8128512*A^2I+139591457711/23708160*ABI+96247214107/28449792*B^2I+177811057747/35562240*ACI+138791552789/35562240*BCI+1798251669/702464*C^2I+15561767153/3556224*ADI+486989759069/142248960*BDI+15067864333/5268480*CDI+1348101763/658560*D^2I+926516417683/237081600*AEI+272269637417/88905600*BEI+963870841/376320*CEI+974923701/439040*DEI+1302893/768*E^2I+719660752261/203212800*AFI+5646525389/2032128*BFI+30635479417/13171200*CFI+26577721169/13171200*DFI+3439289/1920*EFI+1845543/1280*F^2I+65940581957/20321280*AGI+517878924209/203212800*BGI+18744709693/8780800*CGI+8135246289/4390400*DGI+631913/384*EGI+712873/480*FGI+119875/96*G^2I+846491581/282240*AHI+670654454329/284497920*BHI+104091385229/52684800*CHI+235392399/137200*DHI+2926511/1920*EHI+5283859/3840*FHI+321825/256*GHI+52675/48*H^2I+43509745051/16257024*AI^2+59633224631/28449792*BI^2+2462873319/1404928*CI^2+500552257/329280*DI^2+2072339/1536*EI^2+389481/320*FI^2+1707125/1536*GI^2+49175/48*HI^2+14175/16*I^3
        +2610459129/49280*A^2+9515185733/147840*AB+491481463/13440*B^2+19473122233/354816*AC+6953958869/161280*BC+299166571/10752*C^2+239438628925/4967424*AD+1338931253/35280*BD+23979875821/752640*CD+560787183/25088*D^2+357484095329/8279040*AE+5125511809/150528*BE+2153808151/75264*CE+3124551441/125440*DE+58325619/3136*E^2+69605365553/1774080*AF+49957685657/1612800*BF+14006763089/537600*CF+2033250307/89600*DF+316924751/15680*EF+5076027/320*F^2+55939912601/1552320*AG+80372842099/2822400*BG+2818617769/117600*CG+2619903121/125440*DG+583615241/31360*EG+5390121/320*FG+8819811/640*G^2+92118217689/2759680*AH+94607425/3584*BH+83652841679/3763200*CH+12154183701/627200*DH+1083352149/62720*EH+5004153/320*FH+18333777/1280*GH+777315/64*H^2+22092061505/709632*AI+55621863319/2257920*BI+223142029/10752*CI+2270236243/125440*DI+202410923/12544*EI+3740715/256*FI+8567251/640*GI+791945/64*HI+86625/8*I^2+25901005217/887040*AJ+7456563257/322560*BJ+2932648169/150528*CJ+426359263/25088*DJ+95056049/6272*EJ+17570553/1280*FJ+16099237/1280*GJ+744205/64*HJ+86625/8*IJ+155925/16*J^2
        +61154055/176*A+24343173/88*B+1873197/8*C+6559335/32*D+41071455/224*E+9324585/56*F+1223397/8*G+1133433/8*H+1057455/8*I+496125/4*J+467775/4*K
        6081075/4
       
1/24576 146408989/93117911040 8583072000373/715145556787200 498197701343978243/11174149324800000000 24496866073770929/232794777600000000 441238531261177743389/2504052369358848000000 2745921459415330190117/12520261846794240000000 2038875590064298429/9936715751424000000 3225051882695327/22712493146112000 34338052012597/495709175808000 561745903/26227998720 13/4096
Average 8492672176875167143532237/1032921602360524800000000

Level 13:
1/53248 721853003/807021895680 281503678355437/37187568952934400 88715784400749967651/2905278824448000000000 37798534825398160559/484213137408000000000 1624568664901010792829401/11393438280582758400000000 597201188730712614008399/3038250208155402240000000 271840207150663884092677/1302107232066600960000000 7206364035383776819/42276208833331200000 62218993549362983401/595445861980569600000 673746715476949/14768002529280000 7434422179/584509685760 7/4096
Average 719162129942703833322750483827/82716361917030825984000000000

Level 14:
1/114688 52210603817/101684758855680 2498623818382213/520625965341081600 1417146972089835739427/67789839237120000000000 2123479359236610436013/36976275947520000000000 40108924042491076211600383/354462524284796928000000000 260340554695322607053464361/1531278104910322728960000000 919767658082469049944284141/4593834314730968186880000000 6103543220301401334681989/32813102248078344192000000 93543322656237650447677397/693241890352258351104000000 56767708423524073013/764155522875064320000 3030035245383284053/103210616076632064000 7549263763/1013150121984 15/16384
Average 91008379133205771959239329083567887/9935889393473742817198080000000000

Level 15:
1/245760 38858238427/130737547100160 286654638384296387/93712673761394688000 524317697587972030903487/36606513188044800000000000 27470205268281938382581/653687735500800000000000 16481168623071937881494529653/186092825249518387200000000000 230461070660662046085895546481/1607842010155838865408000000000 332937408216669435208270547753/1808822261425318723584000000000 78516278535000936213639608359/413445088325787136819200000000 580641756417032863205816631661/3695512538570115671654400000000 73963260275885859687239/720489492996489216000000 5964476097641414388649279/116228055029297283072000000 1775418549993086407/95838429214015488000 38251327319/8865063567360 1/2048
Average 430482610611138238966092369119663819653/44756213772902474520068751360000000000

Level 16:
1/524288 3091326263/17827847331840 3526412638168580179/1799283336218778009600 57571002542002385913017683/5857042110087168000000000000 76937532337595104977215977/2510160904323072000000000000 6952503351845960447728646209399547/101294046639817848520704000000000000 3203221648348252829113850434898989/27011745770618092938854400000000000 6835521222667500019563484155780187/41675264903239343391375360000000000 370126232255892789346228568939/2003618504963429970739200000000 144927937824466983151671672077843987/852297534975441205399922933760000000 33558129078692645831227750026007/264229146507763270523289600000000 2500003356918612233697998617073/33093484966208449523220480000000 238954724342083526719935457/6907267270312524251136000000 23502514183211758601/2044553156565663744000 2631526038067/1063807628083200 17/65536
Average 6497374168950664548739248365120838691744731463/645133967808125428722079009603584000000000000

Level 17:
1/1114112 1863464628559/18288315160068096 66188299948170023399/52436257226947244851200 40385906698116373280509362629/5974182952288911360000000000000 22218887681869543934889582691/995697158714818560000000000000 424449809305769911205898210910751441/8035994366758882649309184000000000000 24876635804464439104365649844381920603/257151819736284244777893888000000000000 254889920303821362019123288140992935447/1785368348454773470886520422400000000000 12628611959473131391838847896596033/72872177487949937587204915200000000 58363586040687553029368958978065141740241/334697241984855761360549736087552000000000 19281842844302909706866894892029978789/132816365867006254508154657177600000000 812310853734815176311723010378115121977/8210753734908150725124253600972800000000 9146780377555490265092837488615763/169258992680027844261477089280000000 5005907400871473007812936173/219190614711250769569382400000 1603556631633520054331/228095461529356861440000 45197664707939/32150630537625600 9/65536
Average 166290574652981280453599725306444126898090108578925147/15808672411474853405611610649250832056320000000000000

Level 18:
1/2359296 694495381956869/11521638550842900480 899692116487219245181/1101161401765892141875200 1879518068144214766515427683761/403257349279501516800000000000000 3080666257483704245374332616451/189768164366824243200000000000000 2456945008113422743031710861837506101207/60752117412697152828777431040000000000000 91080191659578309752096300063073263104753/1166440654323785334312526675968000000000000 10767379843382292204897474558723786727049/88218200747177042090863362048000000000000 1571338083053592507378366368657973023/9937115111993173307346124800000000000 150567368160565984436975415678164136948210187/877103654730901421683087690747084800000000000 51844898966907878597657386291328834640250673/331350269565007203746944238726676480000000000 2113923747234756480284869396073440195693418041/17752963295469007171834656165879349248000000000 122974361191777525381361831238699894921193/1646842606258720516867778865109401600000000 53638702451676643940708540885432209/1421775538512233891796407549952000000 438097493609107713150601502789/29590732986018853891866624000000 2370493538295982547528981/558377689823865596805120000 866632158911641/1093121438279270400 19/262144
Average 117953595674786257235074513743736834110025844842732606522003/10760647137042703216131711136732056364095897600000000000000

Level 19:
1/4980736 100698513393601/2806552980333527040 33355049570040529612889/62766199900655852086886400 89776097358043250445541654564243/27861416859311013888000000000000000 1551183296331865733947880366074273/131346679479609065472000000000000000 5609850967158477292836887020953886006687511/181800711357496229840116462387200000000000000 129324087609038853856484788220979457810492219/2068488093667512659514213972049920000000000000 1023106037973564506400479651375220528853958311/9942381232639231178807881223897088000000000000 260725704328741809231464741335068596365103039/1846442228918714361778606513009459200000000000 3025380262497952547074280602709117492123858052197/18478118114946846430881944998506984898560000000000 167452455164873426620947884458071687572764193621/1038783095086297583746670188408130764800000000000 1532878826573189559170359872499488426931267748643817/11395471800932720349787912239636502837985280000000000 16627125840588929095780522541961477652316281097/176182309847444655995484110074775863296000000000 114411161101793953892359613218594332317111/2086000634594379321365853229138575360000000 435334952633006554677409579961743967/16883584519832777465082339655680000000 3077314432152368625761405346703/325779879350836143799979212800000 1625684306805583731334567/641265755779941645397524480 16601116665837019/37384753189151047680 5/131072
Average 385505462910834226878588650249250257267223622679349180575265679637989/33817481022899363098547713491454933373339377185325056000000000000000

Level 20:
1/10485760 6717316440139801/312730189237164441600 83030977749805266867149/239109332954879436521472000 268497750485095260966132636325091/120186504098988687360000000000000000 948419078627571458955250878853195051/110331210762871614996480000000000000000 8967245050904920961964072101144381134228053239/381781493850742082664244571013120000000000000000 1296764465116125138579904485960474278406543756647/26062949980210659509879096047828992000000000000000 2117572418960017128850345572609579897601876060283/24675182513731910107405014310217318400000000000000 192266800940925669522355920814346793146559405323/1551011472291720063894029470927945728000000000000 20403860372598931451558958303139160348546663084085410263/133963513545962337109058580939945100978421760000000000000 1357788286525232148411215324611713203972012926233343/8469137469350637947487558124315701411840000000000000 377279426904890680015276180487946670390464098343589748647/2607283948053406416031474320428831849331032064000000000000 635809025280855033102282232590081611428130258587721/5702876714812645612645740140377485931551129600000000 191040955726512532575574891238006446244657588669/2630989160388506862865896043783319558553600000000 114883914604532350588950213778786800005024059/2920400888432131049912194520794005504000000000 5509482737976433579448983739963688824627933/318524354864403593034046213356886425600000000 38907695737165436536209635128648697/6547451619664360187837804445696000000 1304478692866158075393944821/870289239987063661610926080000 3344738931830159/13529720201787998208 21/1048576
Average 15526225346871553628852438219649810173292958697066691663132713538499126698241/1312076330012026893013409084303722010768184893962902369730560000000000000000

评分

参与人数 1威望 +12 金币 +12 贡献 +12 经验 +12 鲜花 +12 收起 理由
KeyTo9_Fans + 12 + 12 + 12 + 12 + 12 我还没开始写，你竟然如此神速算出这么多了

查看全部评分

毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-27 07:20:26 | 显示全部楼层
迭代式可以另外写成
$f_{n,h}(x_1,x_2,...,x_h) = f_{n-1,h} (x_1,x_2,...,x_h) \frac{\sum_{t=1}^h x_t}{h+1} +\sum_{t=0}^{h-1} \frac{(t+1)f_{n-1,h-1}(x_1,x_2,...,x_t, x_{t+2},...,x_h)}{h}$, 而且$n\lt h$时，$f_{n,h}=0$, 而且$f_{n,0}=0$
所以马上迭代可以得出$f_{n,1}(x_1)=f_{n-1,1}(x_1) \frac{x_1}2$, 所以$f_{n,1}(x_1)=(\frac{x_1}2)^{n-1}$, 所以n次试验后还是只有一个人的概率为$\frac1{n2^{n-1}}$
而对h比较小的情况都可以比较容易计算出来，比如
$f_{30,2}(x_1,x_2)=22876658237233/2046980738154938499072x_1^28 + 2541630947305/113721152119718805504x_2x_1^27 + 2540062277609/56860576059859402752x_2^2x_1^26 + 2532858560489/28430288029929701376x_2^3x_1^25 + 2508920094697/14215144014964850688x_2^4x_1^24 + 2447591636969/7107572007482425344x_2^5x_1^23 + 2321571839977/3553786003741212672x_2^6x_1^22 + 2108222840809/1776893001870606336x_2^7x_1^21 + 1804850022377/888446500935303168x_2^8x_1^20 + 4311895963067/1332669751402954752x_2^9x_1^19 + 1053702648937/222111625233825792x_2^10x_1^18 + 705873870089/111055812616912896x_2^11x_1^17 + 429969342257/55527906308456448x_2^12x_1^16 + 237450723259/27763953154228224x_2^13x_1^15 + 237450723259/27763953154228224x_2^14x_1^14 + 429969342257/55527906308456448x_2^15x_1^13 + 705873870089/111055812616912896x_2^16x_1^12 + 1053702648937/222111625233825792x_2^17x_1^11 + 4311895963067/1332669751402954752x_2^18x_1^10 + 1804850022377/888446500935303168x_2^19x_1^9 + 2108222840809/1776893001870606336x_2^20x_1^8 + 2321571839977/3553786003741212672x_2^21x_1^7 + 2447591636969/7107572007482425344x_2^22x_1^6 + 2508920094697/14215144014964850688x_2^23x_1^5 + 2532858560489/28430288029929701376x_2^24x_1^4 + 2540062277609/56860576059859402752x_2^25x_1^3 + 2541630947305/113721152119718805504x_2^26x_1^2 + 22876658237233/2046980738154938499072x_2^27x_1 + 3/536870912x_2^28$
而n次尝试后只有两人的概率为
$3/4, 5/12, 31/144, 481/4320, 1511/25920, 1123/36288, 1133/68040, 2133889/235146240, 35303591/7054387200, 6496405/2327947776, 146408989/93117911040, 721853003/807021895680, 52210603817/101684758855680, 38858238427/130737547100160, 3091326263/17827847331840, 1863464628559/18288315160068096, 694495381956869/11521638550842900480, 100698513393601/2806552980333527040, 6717316440139801/312730189237164441600, 20974007186069533/1622517805336347279360, 2033078119946703031/260066425369625949634560, 7747044278977776727/1631325759136744593162240, 708842463875010695/244698863870511688974336, 3472329061473680008921/1957590910964093511794688000, 33285821273155605689963/30538418211039858783997132800, 3196484217123464894063/4756945913642747233661091840, 174772565732140876609667/420614164995779755397401804800, 3010875178412763609198499/11685297726689477910452187955200, 24479043250136758346740471/152807739502862403444374765568000, 4471438961895272154536275/44786559287020762609514931290112, 192023331828123402046045157/3079075950982677429404151526195200, 3046366705306086219165459058373/78036100901704976770818816279891148800, 17717526808463051462476482525629/723607481088537057329410841868081561600, 6876444454580051974690022065079/446934032437037594232871402330285670400, 22261130072215796787893184908999/2298517881104764770340481497698612019200, 86554639773145124633026021162391/14174193600146049417099635902474774118400, 6231047111436711838508363907645047/1615858070416649633549358492882124249497600, 441196714613084942858043654346259/180914252860045939832798989155223480565760, 35463048801901396499572257554503519/22962193632236600055701410162009134071808000, 4425622377025390277820323084131729487/4518959706824162890962037519883397585331814400, 118015135445412213691088621617003692499/189796307686614841420405575835102698583936204800, 794526104871041982589696888968021343/2010157938552817285979664965741235477613117440, 19355179836206624436737101527448228060601/76948845887801845707301574888574494083030135603200, 43236778750886109779921898706470769605743/269820628437746731700927600258637836395040735232000, 5327065470711929681629778036120855980110509/52129345414172668564619212369968829991521870046822400, 4169283269768491492603053575849176213193791/63915110464333445805315729949266130685257249361756160, 105710671939984690384946588338573449863027/2536313907314819277988719442431195662113382911180800, 13356167921382951322650332140667613102830312023/501094466040374215113675322802246464572416834996168294400, 387286673623728991362785099798175926129563747/22701350016930272505904952709253086585884181591490560000, 350321476849239374763588532979225426381928359/32056078036589792994728498002262178012367850489650348032, 2112440029460789452212326173717770489941268021793/301515698708865170579828637091866015539859958429122979430400, 1275093407929901115242507621306663350129977664973/283674533104790308415341735488797020537501381007281383014400, 30247475953508143117327042222407491788991008963061/10481027486292778763556310437533447811438208919321659519795200, 12197176690593763669374559803696492418868078851639/6578182416321894178928735379713184962734491483898579098009600, 276961244362849151059334591451246766020378433766339/232329442612823262592164881365324759820213631044963452688793600, 139287127325578839653826238016200548946331347325347753/181615244282481270414903747261579583676601284154005716159011225600, 821151094570308422558530069932407670243687048516161069/1663213289744828476431223790711307766301506496989315505877260697600, 1834277800192758536846844622312233812429833357476019/5767804707774973360664706327505906634705459442929248521557114880, 62421243942586712069766019967933824310863393333458498101/304540088570518593443096494092311870312448258586664321938215665664000, 982202457416230233870709420614933972908426705461331985011/7430778161120653680011554455852409635623737509514609455292462242201600, 117877245289069226443481469669518093337489077785862464896371/1382124737968441584482149128788548192226015176769717358684397977049497600, 18558189665587058282268429142151132251173093653773105777745/337060097388432850925324110117465301071892733431582684892066087306264576, 373343050370299087689945006852959736553506673663441644539/10498134262513481652574463562628695008468419853558430800209699563110400, 159349450482186982617736948491907296107567171799692771001983773/6933807717704904361892381693845000479193221944878272374922502367443156992000, 453309208957993540815966083645652707347990368526696024971265331/30508753957901579192326479452918002108450176557464398449659010416749890764800, 508818551258497305045082999698216647152200100964748995416691/52941893685040295412739552534201143259936851317019287648980929757115842560, 117885296318111256570720934957321934864012244826271899433769804161/18954256777118126563669014598294693309922591508519245364088152471642613953331200, 4005647295127009378632013615583640608722016238112497924511802599/994810130036372261328470086675203223923422018829483719666898468973129079193600, 343476048152840959893069470444655850880547674801653110901980186289/131704210693945805901964844083738861602035654231990344625465471218399045484544000, 26427604009574282201162321439797724027679108608063443183535853279/15639301395295028101529483924647457851558798244690700156222520066630799547432960, 11493158146895708082187766534586264998390323313423581820088318428161/10492549481570664308117053760354385358591266458710669741174745281066845514550476800, 419978813784807352297842070049935442875151395905827828543201497924751/591264367277280241362666257513303259154300839743485459800934769171696627942739148800, 1149174591924583289300169303244500026067184943226386440952104908444738827/2493953101175568058067726274191113147112840942038021669440342856366216376662473729638400, 226750960329923103974764643783915005383204434649385509297742342895165413/758296551033111909547619475260811429865390826971020102194698841462700925336562958336000, 13424937287584427702145823557746436838794541710061913773042566209126188793/69156645454219806150742896143786002403723643419757033320156534341398324390694541800243200, 42398527794047998411323573825026184809953191442211282014164090055733064769/336319686314205794122560189667675085373898139578186835725392830165537114405272403281182720, 294938554514419520090268851193076123715325188485968802737456544835124643/3601347510022796180988947968765088363741255716083931411353428070743787227644936637644800, 10394716982379977739545457223756705514557517339042380019697081712451325697/195314299574777684898423829824158282856196665229184709793967411784445714565179023713894400, 141367225500872621513469182468905513335385989620205046714479093336946598473/4086180741103901565638077492373837759754640759400048533847476114964061659982034838224896000, 217304141079792476182414620742153031503329090629651538519799572440085968098079/9659338998618477326618113936532484575635034309708812329356184178068005214278172082219184553984, 869292753515289797738996903425667808718790361916589577008016293615078398023922669/59404934841503635558701400709674780140155461004709195825540532695118232067810758305647985007001600, 1717531605744410360635635083389478316588313496630536848988455754354879369295068973/180388155799200064074593277764744149449984265733812070250726739525419997376644863635443271545651200, 31299907071900218897971026445896657679496819292586977817490723908705536712400282191/5050868362377601794088611777412836184599559440546737967020348706711759926546056181792411603278233600, 3806754923569788070504228323399898200629203443703852725561549581203559078351792993/943568814949661873620949452923277089430686928453786213619186022132966140123988517477703266546483200, 54344052618703956289696166939171458590582586143095209848129162096316517514301613993/20684508531641607347220029183690662470264862470810450722083332798914826365855277696864161803901337600, 214866080926726252083920363665136488583795274378194327079237122478119505136699430857/125550156436243244595916921324261928017189048950733200894505810709692318174144825090268516995773235200, 30683754271233489125151535288058674357227735228922288901086932536406227863537961/27516837325301704689344737286147412627963701552615400546721028063199660616776335130830329557811200, 55618088913192502763681677335402778695983466975130948962715593750664969751740158365/76530359328746417252854756893780778571619255712370830091033839241636944100824736188198873427720798208, 2365807038708269681643405811853124264897142108207661676021322849554129541724538712828069/4993605946200703725748772887319195801798156435232196663439958010516810602578814036279976491158782083072000, 1743442550049309641060011406008600467181694819066702759814306064998734704691036735197/5643543729561152993580952965052742398952213556956407058780876539456405425169797283922973928160074137600, 9256655801374931608739126802876252957056848904251166546849608775226508061693870878057973/45941174705046474276888710563336601376543039204136209303647613696754657543725089133775783718660795164262400, 5860302328688239883089190420717746109476383007768745900617447034853291200216586721769111/44582913887679882863485044337985780118453940653753051811191945117894085059823651715907630113065606437666816, 28053455147941361649759906475266552733647037275384119733409357031222102043869587473860911/327065081069868027250332531303641779224870220196627487272531959501720395079871742556866589278993262628044800, 264759291857382731049897050412902587080661392902422783564415396842409102553457523174201863/4729328954586389559339914752974918362441454002352215793867135617344844010688489108821466966088225000521728000, 17738976089151435365104180865025447153088830155088668257513572834531253403779405903937081/485378497970708402142780724647425831934780805504569515686363918622233990570660724326413714940633618474598400, 57525628103894465056530448854456637470052052058655852991541497265019848592917947736293286079/2410583772321726208401906190888975651720895392457894042704757765445462890770129421294701073881162802792245493760, 55228426744890376132798337517678610636716441713442380162540491962510893335680047508448823064257/3543558145312937526350802100606794208029716226913104242775993915204830449432090249303210578605309320104600875827200, 3527006544070378056038908151036028400314448026584640808012216999447925127405311038007504480569/346424216312686782530015873561625371884406095916126385814506317582565715431642792640899783359373557627079150796800, 1264322628789163238654581851272517474697987587500754822546912092526546373163545264536984292781/190062250705444344140708372610649280781368455667510789220394593446868099434504039381769098297573835521906442240000, 1102266169752651274876250336282878204001690823714720997256330303659098066951135120239365191200497/253555532246109112854891358469949126086511152644688972100440870731405695977876941657867921955423051438271776371507200, 57158275695901716158589792225933433296529780752212801637706531045441984626778317541215746099850109/20115405558191322953154714438615964002863218109811991786634975744691518547578237371524188475130228747436227592139571200, 679218535878072604417972951902089523297666389232376275013785472046858414007981321736517084987758889/365627077498889340736753338913666639816749082113641497768835735594687013600098549870645543459720040173987901527713382400, 45417393373970549098858199347141034064760679726496049564545340505224419689363530500770466106523/37389347883545002756411639956822305859294830456521473901802265965330397127265880774180900013579934754372054354493440, 399259822519396731265204721252689123028743446447607436151433288023973123542819906261760233584027604061/502571037434836984721791862215876326729931465596205404205890574744697058693953643094923692464633364311881551918093303808000, 26101794801999407946521269456985493150728997931199220117672234351060842516480570759234027159081446027331/50228385398487422073052226686603868311465721904443614397491578013055723180327138387029802178322500238941761960271153620582400, 284144216407687599224893813056318301171542375833389731465664500352686421636271509935850233249178434713/835751177322955758109566906821472202362663110684553194152747073511864355219003287488936683560103803478141488142600057651200, 730938516408180437035841389422899083911122781231294232023591609987077693308745825319390674292796175194631/3285505028292003676280329424096571521928101220723115516853279295389841153236945723776507890411480072233269818186189346638397440, 1868937745740297460488979887265037261787387207295879321467067027349335616996817327148413832713240424820581/12835844017341519738872971585180153974557814804975612592724281118189701996517099781062342654295746518760803232340309633820262400, 1787876660846170745716890390888531262497790622062517509618700672314055212011820991775272488838221575117499909/18758669185343392418381499902341853594246635064985788117652770834154378774909990108609680764777926698131859580977338221997326336000, 2156882699076306255809731489112265890436793713082869472540030902073082421121441428170245959033338155775691587/34566369858526045634187570175777755516354866669155630120354631473844946185229396793019103448772606429522027417595395569546061414400, 1843656275251981391296493817574645252404230433198593603479662652164816956012478734748031859626149155631887483/45123126058562378490034044310542313282160542165411268508462937842884078398556212570346559366911267312051727682969124486731750440960, 2673074784867256927354914827764345412247861642958792920612642138276565398784825417674129359184316390038763034759/99897075812915241195895778098114900915281134981710498118327744840736016830109753870024382859643556694109620796907971924911434445619200, 45071101018309497968827687694094165484312402273370296244044733607293765163958268103216717107423586927884444592391/2571544080603431047558865513622441642915946635980805725755662592997011013884760760912240565225663169093531528901050374067074989922713600, 2770046574628261145830694097738655079637148257237348015430949981841037455278948067217926754072129381872605484918341/241251438088190307488088304106947485710403940980830852955761503790377480644715055596108884606039189416274734224532752198661114186170368000, 156920374365230865987123914914745984820535921164542685324059300562974245510210414695918502285528691525674101064339/20858509429736702361703038466886391311110700984181152007106833122745680314126916608061091141093574637730088822394384537921507512245288960, 821910135749622388929216813561276293036824992320975992934636608519163621285519018636198048915483650219950380309849/166718795279320356872818365364801117278461263234004848670987277106656267182205055647384843635871544843535550542039696843438984897809612800, 777441138220252508990646307666406094259018498899013918743273680931312843069608681345297698749404761730925747575794661/240613695156200656588293705459569120182960786215107613135774099467314221913271634927402648945863224950339538405365291705892479588669535027200, 384053990211095873990547577275044200337234237227627041577626308376879703709870743493096979021990578998450287706073/181332266235666122175800969048779057073441403469189902471066438207749402532425832773743370269369238432865640043645617557970186913867366400, 772174896260605728589801483409364537345209555230876859738788954578971534688382498068015674541672558823400596506637/556117939498152519079261260692378552040741416521820369343083434824300975146236444362255737163012076156408794465405142617625145428357939200$

浮点结果为:
0.75000000000000000000000000000000000000
0.41666666666666666666666666666666666667
0.21527777777777777777777777777777777778
0.11134259259259259259259259259259259259
0.058294753086419753086419753086419753087
0.030946869488536155202821869488536155203
0.016651969429747207524985302763080540858
0.0090747315372765475646134082348074117621
0.0050044872784981238342006517589507987313
0.0027906145777730711429842659838087364379
0.0015722967511278053687747332008877504991
0.00089446520207703120940432325891859499541
0.00051345555031606954019151637110571941935
0.00029722324832383553477475474461028831128
0.00017339873992969325694407123954336156631
0.00010189372898755652484232797696845556953
6.0277483874553683157240377123355789585 E-5
3.5879783527775813447904402345231427088 E-5
2.1479590622591303087136679530235161455 E-5
1.2926827130702352979005508324490224054 E-5
7.8175339898533215358346056701232104773 E-6
4.7489253667381015466252788538935013069 E-6
2.8967950756408570688427682004200384627 E-6
1.7737766568213138273648289614743756272 E-6
1.0899654672068948499947329653737226157 E-6
6.7196143810592103711259713642930764763 E-7
4.1551754619080520574304384621287706311 E-7
2.5766354001711554593793363651163094677 E-7
1.6019504856086308041987672808044514298 E-7
9.9838858645948817097390362944375614436 E-8
6.2363947783373046758000691015024529970 E-8
3.9037915402043459608881930549469782859 E-8
2.4484996730285631061982824630239888339 E-8
1.5385815255741975562982274765836347109 E-8
9.6849932102839058568405667685850351547 E-9
6.1064948183192075649572441183141620701 E-9
3.8561846646779032462124966589661485744 E-9
2.4387062248456000897851448221478035297 E-9
1.5444103194093293730820054416033471056 E-9
9.7934539454781545825968430053218665196 E-10
6.2179890053643593920866500145814959093 E-10
3.9525556158191679153900061208888888303 E-10
2.5153307516045362412980939207744969231 E-10
1.6024267307220263145909273296972212991 E-10
1.0218937967449776326328252927298656003 E-10
6.5231574184559642879146981200952112840 E-11
4.1678859874210113619013780192528322096 E-11
2.6653992064456000824007883881645961120 E-11
1.7060072345252476949045218162607928396 E-11
1.0928394810162730166025581637383532680 E-11
7.0060697950606558197924053860327649285 E-12
4.4949167412884307229003533031739857014 E-12
2.8859265938445612714298111395077285866 E-12
1.8541864482702590722786390852018288659 E-12
1.1921056636132198386172829303237245329 E-12
7.6693521998039990733282939844074324576 E-13
4.9371364432537097234698495198993321167 E-13
3.1802009484131124815990844334336572142 E-13
2.0496889009123865885045617408485256302 E-13
1.3218029607656890387473034793418943885 E-13
8.5286980292628803526863005862575080161 E-14
5.5058993364617516622181307808728077804 E-14
3.5562800116152509758798100176772994212 E-14
2.2981521404941954772189167792385315355 E-14
1.4858332450532259456377144919439564907 E-14
9.6108868769511607694125704303349040351 E-15
6.2194628733965564337896056220505361517 E-15
4.0265445376803258411298857966309582563 E-15
2.6079352083207914159675632198879571232 E-15
1.6898199824657665164083155985938284214 E-15
1.0953637309104459296427159974540043819 E-15
7.1030631478567259219367903743130730231 E-16
4.6078436333983181564271983043636341785 E-16
2.9902675941357638716827624343031815929 E-16
1.9412360445492463836767188790543926381 E-16
1.2606614931971982113573856154300512337 E-16
8.1896721628108748550520467434398101931 E-17
5.3220460585889027074051291741523343705 E-17
3.4596420094398854027634040190461049188 E-17
2.2496792079755385385993257024599771903 E-17
1.4633342429120095365463042003011944851 E-17
9.5213102996423382626613893960792869976 E-18
6.1969358190056596875653257332472823557 E-18
4.0344221462775593345250483483164412589 E-18
2.6272827577978227946453780508728735212 E-18
1.7113963616273097302879942016563260661 E-18
1.1150901503865710409267674983475042576 E-18
7.2674543019297147597148534154224268833 E-19
4.7376726641962045319448129514392040033 E-19
3.0892691429271184057957880168981421207 E-19
2.0148931455943204181383318107545406105 E-19
1.3144727021325731805501510288939121670 E-19
8.5773311709630504914342035673687734186 E-20
5.5982422538111994263910246509762637930 E-20
3.6546687097420508781119197369070585813 E-20
2.3863774727268371943295884159771663174 E-20
1.5585585019379740334443848545601089258 E-20
1.0181177810291582861438418535379085533 E-20
6.6521501460518411296694813101916019605 E-21
4.3472377036631031195895537955761539242 E-21
2.8415174394843821150179077919061523246 E-21
1.8576811666256743668385573036391467471 E-21
1.2147147769313911303869059999416615182 E-21
7.9443460283197175966806823991059893699 E-22
5.1966223072711864566897963724453053129 E-22
3.3998661816767858791474166417191630179 E-22
2.2247371716492691781932421445036942677 E-22
1.4560302721155848342034073020971191567 E-22
9.5309355007074945429111473967440003090 E-23
6.2398299500469399356757328851251835429 E-23
4.0858345515761020119171524214972026641 E-23
2.6758288599691595538180003261954575312 E-23
1.7526863085206474320634914157610200131 E-23
1.1481989896431875055970625367399567366 E-23
7.5230866756719958460393059273702887710 E-24
4.9299188755088812228281322147060078634 E-24
3.2310760105136838310958331670440744493 E-24
2.1179572625644301990243346871789759075 E-24
1.3885092377300858011878203533297969735 E-24

n次只有一人的概率极限情况每增加一人，概率减半。而n次余两人的概率好像是每增加一人，概率是原先的2/3.

点评

mathe
是的，我也觉得是这样。  发表于 2021-10-27 15:54
KeyTo9_Fans
n趋于无穷时，n次只有k人的概率好像是每增加1人，概率是原先的k/(k+1)  发表于 2021-10-27 15:43
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-27 16:06:26 | 显示全部楼层
上面Fans提到n趋于无穷时，n次只有k人的概率好像是每增加1人，概率是原先的k/(k+1)， 我也觉得是这样。
为此我计算了$k<=3,n<1000$的情况，结果如下，和猜测的匹配程度还是挺好的。
vf.tgz (20.43 KB, 下载次数: 4)
比如k=1, n=11/n=10 为0.4545
     k=1,n=101/n=100为0.4950
    k=1,n=999/n=998为0.4995

k=2,n=11/n=10为0.5631
k=2,n=101/n=100为0.6537
k=2,n=999/n=998为0.6653

k=3,n=11/n=10为0.6181
k=3,n=101/n=100为0.7286
k=3,n=999/n=998为0.7478
我们可以猜测n次k个人的概率极限为$\frac{c_k}{n^k}(\frac{k}{k+1})^n$, 其中$c_1=2,c_2=30, c_3=600$
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-27 19:26:39 | 显示全部楼层
根据14#,我们有
$f_{n,h}(x_1,x_2,...,x_h) = f_{n-1,h} (x_1,x_2,...,x_h) \frac{\sum_{t=1}^h x_t}{h+1} +\sum_{t=0}^{h-1} \frac{(t+1)f_{n-1,h-1}(x_1,x_2,...,x_t, x_{t+2},...,x_h)}{h}$
由于在n充分大时$f_{n-1,h-1}$要比$f_{n,h}$以远远更加的速度指数衰减，所以对于充分大的n,我们可以认为
$f_{n,h}(x_1,x_2,...,x_h) = f_{n-1,h} (x_1,x_2,...,x_h) \frac{\sum_{t=1}^h x_t}{h+1}$
如果对于任意的初始正函数$g(x_1,x_2,...,x_h)$我们能够证明
$\lim_{n\to\infty}\frac{\int_{0\lt x_1\lt x_2\cdots\lt x_h} g(x_1,x_2,...,x_h) (x_1+x_2+...+x_h)^n dx_1dx_2\cdots\dx_h}{\int_{0\lt x_1\lt x_2\cdots\lt x_h}g(x_1,x_2,...,x_h)(x_1+x_2+...+x_h)^{n+1}dx_1dx_2\cdots\dx_h}=\frac{1}{h}$,和g的初始选择无关，那么就可以证明上述每增加一个人概率会减少$k/{k+1}$的结论。
这个极限的几何意义是当n越来越大时，$x_1+x_2+...+x_h$绝对值越大的部分起的作用越大，也就是最终极限应该收敛到
$\lim_{x_i\to 1} \frac{g(x_1,x_1,...,x_1) (x_1+x_1+...+x_1)^n }{g(x_1,x_1,...,x_1)(x_1+x_1+...+x_1)^{n+1}}=1/h$
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-28 08:42:07 | 显示全部楼层
如果我们选择上面g=1分别进行计算，可以得到h=1时，结果为
\(\int_0^1 (\frac{x_1}2)^n dx_1=\frac1{(n+1)2^n}\)
而h=2结果为
\(\int_0^1 dx_2 \int_0^{x_2}(\frac{x_1+x_2}3)^ndx_1=\frac{2^{n+1}-1}{(n+1)(n+2)3^n}\)
h=3的结果为
\(\int_0^1 dx_3\int_0^{x_3}dx_2 \int_0^{x_2}(\frac{x_1+x_2+x_3}4)^ndx_1=\frac{3^{n+2}-2^{n+3}+1}{2(n+1)(n+2)(n+3)4^n}\)
这个结果和前面的预测基本一致（除了乘数$c_k$会不同,这时因为g不是1，g(1,1,...,1)要远远大于1)
而一般的结果是近似于\(\frac{h^{n+h-1}}{(h-1)!(n+1)(n+2)...(n+h)(h+1)^n}\)
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 1 反对 0

使用道具 举报

mathe
发表于 2021-10-28 09:55:26 来自手机 | 显示全部楼层
现在可以算出前面c2=30,c3=600估计的不准。应该是c2=27,c3=576
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-28 12:53:36 | 显示全部楼层
现在假设$u_{m,h}=f_{m,h}(1,1,...,1)$
根据17#的结论，我们知道对于充分大的n,n次后正好h个人的概率接近
\(\int_0^1 dx_h\int_0^{x_{h-1}} dx_{h-2}...\int_0^{x_2}f_{m,h}(x_1,x_2,...,x_h)(\frac{x_1+x_2+...+x_h}{h+1})^{n-m}dx_1\approx \frac{u_{m,h}h^{n-m+h-1}}{(h-1)!(n-m+1)(n-m+2)...(n-m+h)(h+1)^{n-m}}\)
即
\(\frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}(\frac{h}{h+1})^n\frac1{(n-m+1)(n-m+2)...(n-m+h)}\approx \frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}(\frac{h}{h+1})^n\frac1{n^h}\)
也就是我们前面求的\(c_h \approx \frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}\)
另外16#第一行公式中将$x_i=1$代入可以得出递推式\(u_{m,h}=\frac{h u_{m-1,h}}{h+1}+\frac{(h+1)u_{m-1,h-1}}2\)而且我们已经知道$u_{1,1}=1$,由此可以轻松计算$c_h$
设\(\lambda_{m,h}=\frac{(h+1)^m u_{m,h}}{(h-1)!h^{m+1-h}}, \lambda_{m,1}=2\)
递推式变化为\(\lambda_{m,h}=\lambda_{m-1,h}+\frac{(h+1)}2 (\frac h{h-1})^h (\frac{h^2-1}{h^2})^m \lambda_{m-1,h-1}\)
特别的\(\lambda_{h,h}=\frac{(h+1)}2 (\frac h{h-1})^h (\frac{h^2-1}{h^2})^h \lambda_{h-1,h-1}\)
而且\(c_h = \lim_{m\to\infty}\lambda_{m,h}\)
比如让h=2代入，我们可以化简得到\(\lambda_{2,2}=\frac{27}4, \lambda_{m,2}=\lambda_{m-1,2}+12(\frac34)^m\),可以得到\(c_2=\lim_{m\to\infty} \lambda_{m,2}=27\)
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

mathe
发表于 2021-10-28 14:31:41 | 显示全部楼层
计算结果$c_3$好像是864而不是576，和前面的模拟结果有偏差，看来还有问题
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
回复 支持 反对

使用道具 举报

123下一页
返回列表 发新帖
高级模式
B Color Image Link Quote Code Smilies
您需要登录后才可以回帖 登录 | 欢迎注册

本版积分规则

小黑屋|手机版|数学研发网 ( 苏ICP备07505100号 )

GMT+8, 2024-5-9 05:42 , Processed in 0.070868 second(s), 23 queries .

Powered by Discuz! X3.5

© 2001-2024 Discuz! Team.

快速回复 返回顶部 返回列表