{1,2,...,n}的和能被 m 整除的 m 元子集

mathe

对于任意M,我们同样可以假设从1，2,...,kM中选择u个数，这u个数的和模M为h的不同组合情况为$C_M(u,kM,h)$,
我们可以先查看从0,1,2,...,M-1中选择u个数（每个数最多可以重复k次),这u个数的和模M为h的所有模式，
其中比如某个模式中有$a_0$个0，$a_1$个1，..., $a_{M-1}$个M-1. 于是$a_0+a_1+...+a_{M-1}=u$, 而且这种模式回归到原题对
$C_M(u,kM,h)$贡献了$C_k^{a_0}C_k^{a_1}...C_k^{a_{M-1}}$种情况，这时k的一个M次多项式。所以所有这些模式得到的计数之和$C_M(u,kM,h)$也是k的M次多项式。
同样，如果我们查看$C_M(u,kM+1,h)$， 那么我们同样根据其中0,1,...,M-1的数目得出所有模式，区别是$a_1$最大有k+1个选择，得出各个模式的贡献为$C_k^{a_0}C_{k+1}^{a_1}C_k^{a_2}...C_k^{a_{M-1}}$这同样是k的一个M次多项式。
所以$C_M(u,n,h)$,特别是$F(n,M)=C_M(M,n,0)$可以根据n%M表示为M个$\lfloor\frac nM\rfloor$的M次多项式。

王守恩

TSC999 发表于 2021-8-1 08:40
对于 m 为奇素数，@王守恩 拟出了一个与陆教授公式等价的公式：\begin{equation}F(n, m)=\left\lfloor{\fra ...

我们可以这样说吗？m 为奇素数，有：\(F(n,m)=\lfloor\frac{C_{n}^m}{m}\rfloor+\lfloor\frac{n}{m}\rfloor-\lfloor\frac{n}{m^2}\rfloor\)，倒回去：

若 \(\lfloor\frac{C_{n}^m}{m}\rfloor+\lfloor\frac{n}{m}\rfloor-\lfloor\frac{n}{m^2}\rfloor-F(n,m)=0\)  则：m 为奇素数。在这里 \(m<n<4m/3\)

或(从33楼来的)，m 为奇素数，有：\(F(n,m)=\frac{n!/m}{m!(n-m)!}-\frac{(1-m)\lfloor n/m\rfloor}{m}\)，倒回去：

若 \(\frac{n!/m}{m!(n-m)!}-\frac{(1-m)\lfloor n/m\rfloor}{m}-F(n,m)=0\)  则：m 为奇素数。在这里 \(m<n<4m/3\)

mathe

根据楼上的结论可以看出2#的公式(3) (m=2)时的结论是错误的，因为分别去n为奇数和偶数，结果不是[n/2]的多项式。
通过一段简单的Pari/gp代码可以直接计算出m个关于k的多项式 (设n=km+r):

1. getPolys(m)={
   
2.    local(C, P, H, R, r);
   
3.    C=getC(m*(m+1),m);
   
4.    H=matrix(m+1,m+1);
   
5.    for(u=1,m+1,
   
6.       H[u,1]=1;
   
7.       for(v=1,m, H[u,v+1]=(u-1)^v)
   
8.    );
   
9.    R=matrix(m+1,m);
   
10.    for(u=1,m+1,
    
11.       for(v=1,m,
    
12.          R[u,v]=C[(u-1)*m+v]
    
13.       )
    
14.    );
    
15.    R=H^-1*R;
    
16.    P=vector(m);
    
17.    for(u=1,m,
    
18.       P[u]=R[m+1,u];
    
19.       for(v=1,m,
    
20.          if(u==m, P[u]*=k-1,P[u]*=k);
    
21.          P[u]+=R[m+1-v,u]
    
22.       )
    
23.    );
    
24.    P
    
25. }

复制代码

比如
getPolys(2)
%10 = [k^2, k^2 - k]
代表n为奇数时，结果为k^2 (k=(n-1)/2), 而n为偶数时，结果为k^2-k (k=n/2)
同样:
? getPolys(3)
%11 = [3/2*k^3 + 1/2*k, 3/2*k^3 + 3/2*k^2 + k, 3/2*k^3 - 3/2*k^2 + k]
代表n=3k+1时，结果为3/2*k^3 + 1/2*k; n=3k+2时，结果为3/2*k^3 + 3/2*k^2 + k, n=3k时，结果为3/2*k^3 - 3/2*k^2 + k
类似有
? getPolys(2)
%10 = [k^2, k^2 - k]
? getPolys(3)
%11 = [3/2*k^3 + 1/2*k, 3/2*k^3 + 3/2*k^2 + k, 3/2*k^3 - 3/2*k^2 + k]
? getPolys(4)
%12 = [8/3*k^4 - 4/3*k^3 + 1/3*k^2 - 2/3*k, 8/3*k^4 + 4/3*k^3 + 1/3*k^2 - 1/3*k, 8/3*k^4 + 4*k^3 + 7/3*k^2, 8/3*k^4 - 4*k^3 + 7/3*k^2 - k]
? getPolys(5)
%13 = [125/24*k^5 - 125/24*k^4 + 25/24*k^3 + 5/24*k^2 + 3/4*k, 125/24*k^5 - 25/24*k^3 + 5/6*k, 125/24*k^5 + 125/24*k^4 + 25/24*k^3 - 5/24*k^2 + 3/4*k, 125/24*k^5 + 125/12*k^4 + 175/24*k^3 + 25/12*k^2 + k, 125/24*k^5 - 125/12*k^4 + 175/24*k^3 - 25/12*k^2 + k]
? getPolys(6)
%14 = [54/5*k^6 - 81/5*k^5 + 15/2*k^4 - 3/2*k^3 + 6/5*k^2 - 4/5*k, 54/5*k^6 - 27/5*k^5 - 3/2*k^4 + 7/10*k^2 - 3/5*k, 54/5*k^6 + 27/5*k^5 - 3/2*k^4 - 3/2*k^3 + 7/10*k^2 + 1/10*k, 54/5*k^6 + 81/5*k^5 + 15/2*k^4 - 3/10*k^2 - 1/5*k, 54/5*k^6 + 27*k^5 + 51/2*k^4 + 21/2*k^3 + 11/5*k^2, 54/5*k^6 - 27*k^5 + 51/2*k^4 - 12*k^3 + 37/10*k^2 - k]
? getPolys(7)
%15 = [16807/720*k^7 - 16807/360*k^6 + 2401/72*k^5 - 343/36*k^4 + 343/720*k^3 + 77/360*k^2 + 5/6*k, 16807/720*k^7 - 16807/720*k^6 + 2401/720*k^5 + 343/144*k^4 - 49/90*k^3 - 7/180*k^2 + 13/15*k, 16807/720*k^7 - 2401/360*k^5 + 343/720*k^3 + 17/20*k, 16807/720*k^7 + 16807/720*k^6 + 2401/720*k^5 - 343/144*k^4 - 49/90*k^3 + 7/180*k^2 + 13/15*k, 16807/720*k^7 + 16807/360*k^6 + 2401/72*k^5 + 343/36*k^4 + 343/720*k^3 - 77/360*k^2 + 5/6*k, 16807/720*k^7 + 16807/240*k^6 + 12005/144*k^5 + 2401/48*k^4 + 1421/90*k^3 + 49/20*k^2 + k, 16807/720*k^7 - 16807/240*k^6 + 12005/144*k^5 - 2401/48*k^4 + 1421/90*k^3 - 49/20*k^2 + k]
? getPolys(8)
%16 = [16384/315*k^8 - 8192/63*k^7 + 5632/45*k^6 - 512/9*k^5 + 568/45*k^4 - 20/9*k^3 + 127/105*k^2 - 6/7*k, 16384/315*k^8 - 8192/105*k^7 + 512/15*k^6 - 8/5*k^4 - 4/15*k^3 + 143/315*k^2 - 5/7*k, 16384/315*k^8 - 8192/315*k^7 - 512/45*k^6 + 256/45*k^5 + 88/45*k^4 - 44/45*k^3 + 43/105*k^2 - 74/105*k, 16384/315*k^8 + 8192/315*k^7 - 512/45*k^6 - 256/45*k^5 + 88/45*k^4 + 44/45*k^3 + 43/105*k^2 - 31/105*k, 16384/315*k^8 + 8192/105*k^7 + 512/15*k^6 - 8/5*k^4 + 4/15*k^3 + 143/315*k^2 - 2/7*k, 16384/315*k^8 + 8192/63*k^7 + 5632/45*k^6 + 512/9*k^5 + 568/45*k^4 + 20/9*k^3 + 127/105*k^2 - 1/7*k, 16384/315*k^8 + 8192/45*k^7 + 11776/45*k^6 + 1792/9*k^5 + 3928/45*k^4 + 1028/45*k^3 + 421/105*k^2, 16384/315*k^8 - 8192/45*k^7 + 11776/45*k^6 - 1792/9*k^5 + 3928/45*k^4 - 1028/45*k^3 + 421/105*k^2 - k]
? getPolys(9)
%17 = [531441/4480*k^9 - 1594323/4480*k^8 + 137781/320*k^7 - 85293/320*k^6 + 55647/640*k^5 - 8181/640*k^4 + 69/70*k^3 - 897/1120*k^2 + 7/8*k, 531441/4480*k^9 - 531441/2240*k^8 + 373977/2240*k^7 - 6561/160*k^6 - 2673/640*k^5 + 1053/320*k^4 + 789/1120*k^3 - 579/560*k^2 + 25/28*k, 531441/4480*k^9 - 531441/4480*k^8 + 19683/2240*k^7 + 6561/320*k^6 - 2673/640*k^5 - 567/640*k^4 + 339/280*k^3 + 9/1120*k^2 + 31/56*k, 531441/4480*k^9 - 19683/448*k^7 + 3159/640*k^5 + 183/224*k^3 + 39/70*k, 531441/4480*k^9 + 531441/4480*k^8 + 19683/2240*k^7 - 6561/320*k^6 - 2673/640*k^5 + 567/640*k^4 + 339/280*k^3 - 9/1120*k^2 + 31/56*k, 531441/4480*k^9 + 531441/2240*k^8 + 373977/2240*k^7 + 6561/160*k^6 - 2673/640*k^5 - 1053/320*k^4 + 789/1120*k^3 + 579/560*k^2 + 25/28*k, 531441/4480*k^9 + 1594323/4480*k^8 + 137781/320*k^7 + 85293/320*k^6 + 55647/640*k^5 + 8181/640*k^4 + 69/70*k^3 + 897/1120*k^2 + 7/8*k, 531441/4480*k^9 + 531441/1120*k^8 + 255879/320*k^7 + 59049/80*k^6 + 259767/640*k^5 + 21627/160*k^4 + 30651/1120*k^3 + 1041/280*k^2 + k, 531441/4480*k^9 - 531441/1120*k^8 + 255879/320*k^7 - 59049/80*k^6 + 259767/640*k^5 - 21627/160*k^4 + 30651/1120*k^3 - 1041/280*k^2 + k]
? getPolys(10)
%18 = [156250/567*k^10 - 78125/81*k^9 + 265625/189*k^8 - 59375/54*k^7 + 106625/216*k^6 - 13625/108*k^5 + 87175/4536*k^4 - 1105/324*k^3 + 104/63*k^2 - 8/9*k, 156250/567*k^10 - 390625/567*k^9 + 125000/189*k^8 - 15625/54*k^7 + 9125/216*k^6 + 1375/216*k^5 - 4175/4536*k^4 - 1375/4536*k^3 + 61/84*k^2 - 7/9*k, 156250/567*k^10 - 78125/189*k^9 + 31250/189*k^8 + 3125/126*k^7 - 5875/216*k^6 + 4075/1134*k^4 - 505/756*k^3 + 347/504*k^2 - 65/84*k, 156250/567*k^10 - 78125/567*k^9 - 15625/189*k^8 + 15625/378*k^7 + 1625/216*k^6 - 1375/216*k^5 - 1025/4536*k^4 + 2875/4536*k^3 + 101/126*k^2 - 103/126*k, 156250/567*k^10 + 78125/567*k^9 - 15625/189*k^8 - 15625/378*k^7 + 1625/216*k^6 + 125/108*k^5 - 1025/4536*k^4 + 925/2268*k^3 + 101/126*k^2 - 1/63*k, 156250/567*k^10 + 78125/189*k^9 + 31250/189*k^8 - 3125/126*k^7 - 5875/216*k^6 - 125/24*k^5 - 7325/4536*k^4 - 565/1512*k^3 + 113/126*k^2 + 1/42*k, 156250/567*k^10 + 390625/567*k^9 + 125000/189*k^8 + 15625/54*k^7 + 9125/216*k^6 - 625/54*k^5 - 3475/567*k^4 - 1675/2268*k^3 + 157/168*k^2 + 1/36*k, 156250/567*k^10 + 78125/81*k^9 + 265625/189*k^8 + 59375/54*k^7 + 106625/216*k^6 + 26125/216*k^5 + 39925/4536*k^4 - 2515/648*k^3 - 109/252*k^2 - 1/9*k, 156250/567*k^10 + 78125/63*k^9 + 453125/189*k^8 + 15625/6*k^7 + 376625/216*k^6 + 8875/12*k^5 + 880975/4536*k^4 + 7225/252*k^3 + 163/63*k^2, 156250/567*k^10 - 78125/63*k^9 + 453125/189*k^8 - 15625/6*k^7 + 376625/216*k^6 - 17875/24*k^5 + 928225/4536*k^4 - 18125/504*k^3 + 1177/252*k^2 - k]
? getPolys(11)
%19 = [2357947691/3628800*k^11 - 2357947691/907200*k^10 + 214358881/48384*k^9 - 253333223/60480*k^8 + 412773713/172800*k^7 - 35914373/43200*k^6 + 120949301/725760*k^5 - 2716571/181440*k^4 - 4961/11200*k^3 + 4609/25200*k^2 + 9/10*k, 2357947691/3628800*k^11 - 2357947691/1209600*k^10 + 214358881/90720*k^9 - 19487171/13440*k^8 + 76177123/172800*k^7 - 2093663/57600*k^6 - 5168273/362880*k^5 + 222277/60480*k^4 - 71753/453600*k^3 - 341/12600*k^2 + 41/45*k, 2357947691/3628800*k^11 - 2357947691/1814400*k^10 + 214358881/241920*k^9 - 19487171/120960*k^8 - 12400927/172800*k^7 + 2737867/86400*k^6 - 161051/145152*k^5 - 368687/362880*k^4 + 31823/302400*k^3 + 319/50400*k^2 + 109/120*k, 2357947691/3628800*k^11 - 2357947691/3628800*k^10 + 19487171/120960*k^8 - 33659659/1209600*k^7 - 2093663/172800*k^6 + 980947/362880*k^5 + 54571/181440*k^4 - 1573/21600*k^3 - 11/6300*k^2 + 191/210*k, 2357947691/3628800*k^11 - 214358881/725760*k^9 + 54918391/1209600*k^7 - 2035099/725760*k^5 + 57959/907200*k^3 + 229/252*k, 2357947691/3628800*k^11 + 2357947691/3628800*k^10 - 19487171/120960*k^8 - 33659659/1209600*k^7 + 2093663/172800*k^6 + 980947/362880*k^5 - 54571/181440*k^4 - 1573/21600*k^3 + 11/6300*k^2 + 191/210*k, 2357947691/3628800*k^11 + 2357947691/1814400*k^10 + 214358881/241920*k^9 + 19487171/120960*k^8 - 12400927/172800*k^7 - 2737867/86400*k^6 - 161051/145152*k^5 + 368687/362880*k^4 + 31823/302400*k^3 - 319/50400*k^2 + 109/120*k, 2357947691/3628800*k^11 + 2357947691/1209600*k^10 + 214358881/90720*k^9 + 19487171/13440*k^8 + 76177123/172800*k^7 + 2093663/57600*k^6 - 5168273/362880*k^5 - 222277/60480*k^4 - 71753/453600*k^3 + 341/12600*k^2 + 41/45*k, 2357947691/3628800*k^11 + 2357947691/907200*k^10 + 214358881/48384*k^9 + 253333223/60480*k^8 + 412773713/172800*k^7 + 35914373/43200*k^6 + 120949301/725760*k^5 + 2716571/181440*k^4 - 4961/11200*k^3 - 4609/25200*k^2 + 9/10*k, 2357947691/3628800*k^11 + 2357947691/725760*k^10 + 214358881/30240*k^9 + 214358881/24192*k^8 + 1209976163/172800*k^7 + 125780831/34560*k^6 + 454793383/362880*k^5 + 10175495/36288*k^4 + 1948463/50400*k^3 + 7381/2520*k^2 + k, 2357947691/3628800*k^11 - 2357947691/725760*k^10 + 214358881/30240*k^9 - 214358881/24192*k^8 + 1209976163/172800*k^7 - 125780831/34560*k^6 + 454793383/362880*k^5 - 10175495/36288*k^4 + 1948463/50400*k^3 - 7381/2520*k^2 + k]
? getPolys(12)
%20 = [2985984/1925*k^12 - 13436928/1925*k^11 + 476928/35*k^10 - 528768/35*k^9 + 1827792/175*k^8 - 115992/25*k^7 + 45914/35*k^6 - 8037/35*k^5 + 31709/1050*k^4 - 2941/350*k^3 + 3779/1155*k^2 - 10/11*k, 2985984/1925*k^12 - 1492992/275*k^11 + 1389312/175*k^10 - 31104/5*k^9 + 479952/175*k^8 - 15336/25*k^7 + 4786/175*k^6 + 59/5*k^5 + 1849/1050*k^4 - 92/25*k^3 + 25657/11550*k^2 - 9/11*k, 2985984/1925*k^12 - 1492992/385*k^11 + 642816/175*k^10 - 10368/7*k^9 + 13392/175*k^8 + 648/5*k^7 - 5042/175*k^6 - 293/35*k^5 + 6859/1050*k^4 - 439/210*k^3 + 4958/5775*k^2 - 98/165*k, 2985984/1925*k^12 - 4478976/1925*k^11 + 20736/25*k^10 + 10368/35*k^9 - 38448/175*k^8 + 216/25*k^7 + 3526/175*k^6 - 159/35*k^5 + 107/150*k^4 - 638/525*k^3 + 2777/11550*k^2 - 61/165*k, 2985984/1925*k^12 - 1492992/1925*k^11 - 20736/35*k^10 + 10368/35*k^9 + 13392/175*k^8 - 6696/175*k^7 + 10/7*k^6 - 5/7*k^5 + 1159/1050*k^4 - 1367/1050*k^3 + 551/2310*k^2 - 853/2310*k, 2985984/1925*k^12 + 1492992/1925*k^11 - 20736/35*k^10 - 10368/35*k^9 + 13392/175*k^8 + 6696/175*k^7 + 10/7*k^6 + 5/7*k^5 + 1159/1050*k^4 - 104/525*k^3 + 551/2310*k^2 - 151/1155*k, 2985984/1925*k^12 + 4478976/1925*k^11 + 20736/25*k^10 - 10368/35*k^9 - 38448/175*k^8 - 216/25*k^7 + 3526/175*k^6 + 159/35*k^5 + 107/150*k^4 - 299/1050*k^3 + 2777/11550*k^2 - 43/330*k, 2985984/1925*k^12 + 1492992/385*k^11 + 642816/175*k^10 + 10368/7*k^9 + 13392/175*k^8 - 648/5*k^7 - 5042/175*k^6 + 293/35*k^5 + 6859/1050*k^4 + 62/105*k^3 - 7409/11550*k^2 - 67/165*k, 2985984/1925*k^12 + 1492992/275*k^11 + 1389312/175*k^10 + 31104/5*k^9 + 479952/175*k^8 + 15336/25*k^7 + 4786/175*k^6 - 59/5*k^5 + 1849/1050*k^4 + 109/50*k^3 + 4166/5775*k^2 - 2/11*k, 2985984/1925*k^12 + 13436928/1925*k^11 + 476928/35*k^10 + 528768/35*k^9 + 1827792/175*k^8 + 115992/25*k^7 + 45914/35*k^6 + 8037/35*k^5 + 31709/1050*k^4 + 1208/175*k^3 + 4093/2310*k^2 - 1/11*k, 2985984/1925*k^12 + 1492992/175*k^11 + 20736*k^10 + 1026432/35*k^9 + 4678992/175*k^8 + 411048/25*k^7 + 242726/35*k^6 + 70163/35*k^5 + 59137/150*k^4 + 18519/350*k^3 + 5737/1155*k^2, 2985984/1925*k^12 - 1492992/175*k^11 + 20736*k^10 - 1026432/35*k^9 + 4678992/175*k^8 - 411048/25*k^7 + 242726/35*k^6 - 70163/35*k^5 + 59137/150*k^4 - 9522/175*k^3 + 14939/2310*k^2 - k]
? getPolys(13)
%21 = [1792160394037/479001600*k^13 - 1792160394037/95800320*k^12 + 1792160394037/43545600*k^11 - 455993473039/8709120*k^10 + 617508155797/14515200*k^9 - 66701673571/2903040*k^8 + 361233558751/43545600*k^7 - 16958065189/8709120*k^6 + 5918153371/21772800*k^5 - 7015021/435456*k^4 - 1828073/2217600*k^3 + 55991/332640*k^2 + 11/12*k, 1792160394037/479001600*k^13 - 1792160394037/119750400*k^12 + 12269405774561/479001600*k^11 - 10604499373/435456*k^10 + 201485488087/14515200*k^9 - 17004848107/3628800*k^8 + 34642008193/43545600*k^7 + 4826809/2177280*k^6 - 140834291/5443200*k^5 + 5191511/1360800*k^4 - 248261/3326400*k^3 - 3601/166320*k^2 + 61/66*k, 1792160394037/479001600*k^13 - 1792160394037/159667200*k^12 + 6479349116903/479001600*k^11 - 116649493103/14515200*k^10 + 30182036677/14515200*k^9 + 62748517/537600*k^8 - 8538625121/43545600*k^7 + 507557531/14515200*k^6 + 33444931/21772800*k^5 - 3534973/3628800*k^4 + 375349/6652800*k^3 + 2509/554400*k^2 + 203/220*k, 1792160394037/479001600*k^13 - 1792160394037/239500800*k^12 + 2343594361433/479001600*k^11 - 10604499373/21772800*k^10 - 10604499373/14515200*k^9 + 1819706993/7257600*k^8 + 584043889/43545600*k^7 - 350129299/21772800*k^6 + 12652523/10886400*k^5 + 1649947/5443200*k^4 - 41743/1247400*k^3 - 533/415800*k^2 + 457/495*k, 1792160394037/479001600*k^13 - 1792160394037/479001600*k^12 - 137858491849/479001600*k^11 + 10604499373/8709120*k^10 - 815730721/4838400*k^9 - 1945204027/14515200*k^8 + 1192221823/43545600*k^7 + 51609727/8709120*k^6 - 29960489/21772800*k^5 - 1052363/10886400*k^4 + 471679/19958400*k^3 + 13/33264*k^2 + 731/792*k, 1792160394037/479001600*k^13 - 137858491849/68428800*k^11 + 815730721/2073600*k^9 - 1501137599/43545600*k^7 + 1056757/777600*k^5 - 9971/475200*k^3 + 853/924*k, 1792160394037/479001600*k^13 + 1792160394037/479001600*k^12 - 137858491849/479001600*k^11 - 10604499373/8709120*k^10 - 815730721/4838400*k^9 + 1945204027/14515200*k^8 + 1192221823/43545600*k^7 - 51609727/8709120*k^6 - 29960489/21772800*k^5 + 1052363/10886400*k^4 + 471679/19958400*k^3 - 13/33264*k^2 + 731/792*k, 1792160394037/479001600*k^13 + 1792160394037/239500800*k^12 + 2343594361433/479001600*k^11 + 10604499373/21772800*k^10 - 10604499373/14515200*k^9 - 1819706993/7257600*k^8 + 584043889/43545600*k^7 + 350129299/21772800*k^6 + 12652523/10886400*k^5 - 1649947/5443200*k^4 - 41743/1247400*k^3 + 533/415800*k^2 + 457/495*k, 1792160394037/479001600*k^13 + 1792160394037/159667200*k^12 + 6479349116903/479001600*k^11 + 116649493103/14515200*k^10 + 30182036677/14515200*k^9 - 62748517/537600*k^8 - 8538625121/43545600*k^7 - 507557531/14515200*k^6 + 33444931/21772800*k^5 + 3534973/3628800*k^4 + 375349/6652800*k^3 - 2509/554400*k^2 + 203/220*k, 1792160394037/479001600*k^13 + 1792160394037/119750400*k^12 + 12269405774561/479001600*k^11 + 10604499373/435456*k^10 + 201485488087/14515200*k^9 + 17004848107/3628800*k^8 + 34642008193/43545600*k^7 - 4826809/2177280*k^6 - 140834291/5443200*k^5 - 5191511/1360800*k^4 - 248261/3326400*k^3 + 3601/166320*k^2 + 61/66*k, 1792160394037/479001600*k^13 + 1792160394037/95800320*k^12 + 1792160394037/43545600*k^11 + 455993473039/8709120*k^10 + 617508155797/14515200*k^9 + 66701673571/2903040*k^8 + 361233558751/43545600*k^7 + 16958065189/8709120*k^6 + 5918153371/21772800*k^5 + 7015021/435456*k^4 - 1828073/2217600*k^3 - 55991/332640*k^2 + 11/12*k, 1792160394037/479001600*k^13 + 1792160394037/79833600*k^12 + 2619311345131/43545600*k^11 + 137858491849/1451520*k^10 + 1425081569587/14515200*k^9 + 18761806583/268800*k^8 + 1518596167153/43545600*k^7 + 3567011851/290304*k^6 + 32815532243/10886400*k^5 + 905183773/1814400*k^4 + 21797113/415800*k^3 + 86021/27720*k^2 + k, 1792160394037/479001600*k^13 - 1792160394037/79833600*k^12 + 2619311345131/43545600*k^11 - 137858491849/1451520*k^10 + 1425081569587/14515200*k^9 - 18761806583/268800*k^8 + 1518596167153/43545600*k^7 - 3567011851/290304*k^6 + 32815532243/10886400*k^5 - 905183773/1814400*k^4 + 21797113/415800*k^3 - 86021/27720*k^2 + k]
? getPolys(14)
%22 = [7909306972/868725*k^14 - 3954653486/78975*k^13 + 8191782221/66825*k^12 - 2138741171/12150*k^11 + 443889677/2700*k^10 - 1694027951/16200*k^9 + 4473670453/97200*k^8 - 1353417289/97200*k^7 + 273927689/97200*k^6 - 36467417/97200*k^5 + 7359751/178200*k^4 - 18613/2700*k^3 + 4943/2574*k^2 - 12/13*k, 7909306972/868725*k^14 - 3954653486/96525*k^13 + 5367029731/66825*k^12 - 40353607/450*k^11 + 1516142663/24300*k^10 - 149061283/5400*k^9 + 721843843/97200*k^8 - 1327753/1350*k^7 - 679483/48600*k^6 + 3087/200*k^5 + 998081/1069200*k^4 - 5747/29700*k^3 + 11887/15444*k^2 - 11/13*k, 7909306972/868725*k^14 - 27682574402/868725*k^13 + 282475249/6075*k^12 - 4802079233/133650*k^11 + 40353607/2700*k^10 - 40353607/16200*k^9 - 43345253/97200*k^8 + 25025623/97200*k^7 - 261709/24300*k^6 - 117649/6075*k^5 + 2030903/356400*k^4 - 33271/118800*k^3 + 3493/4680*k^2 - 725/858*k, 7909306972/868725*k^14 - 3954653486/173745*k^13 + 282475249/13365*k^12 - 40353607/5346*k^11 - 5764801/8100*k^10 + 823543/648*k^9 - 4857223/19440*k^8 - 117649/2430*k^7 + 1370971/48600*k^6 - 4459/1944*k^5 - 105007/71280*k^4 + 1757/5940*k^3 + 112951/128700*k^2 - 1849/2145*k, 7909306972/868725*k^14 - 3954653486/289575*k^13 + 282475249/66825*k^12 + 40353607/14850*k^11 - 40353607/24300*k^10 - 823543/16200*k^9 + 16655737/97200*k^8 - 890771/32400*k^7 + 477799/97200*k^6 - 7889/10800*k^5 - 292579/267300*k^4 + 22897/89100*k^3 + 16916/19305*k^2 - 5546/6435*k, 7909306972/868725*k^14 - 3954653486/868725*k^13 - 282475249/66825*k^12 + 282475249/133650*k^11 + 5764801/8100*k^10 - 5764801/16200*k^9 - 5226977/97200*k^8 + 184877/12150*k^7 + 88837/48600*k^6 + 117649/48600*k^5 - 2891/118800*k^4 - 6713/29700*k^3 + 1471/1716*k^2 - 1831/2145*k, 7909306972/868725*k^14 + 3954653486/868725*k^13 - 282475249/66825*k^12 - 282475249/133650*k^11 + 5764801/8100*k^10 + 5764801/16200*k^9 - 5226977/97200*k^8 - 3747961/97200*k^7 + 88837/48600*k^6 + 103243/24300*k^5 - 2891/118800*k^4 - 29743/118800*k^3 + 1471/1716*k^2 + 31/8580*k, 7909306972/868725*k^14 + 3954653486/289575*k^13 + 282475249/66825*k^12 - 40353607/14850*k^11 - 40353607/24300*k^10 + 823543/16200*k^9 + 16655737/97200*k^8 + 16807/4050*k^7 - 895573/48600*k^6 - 14063/5400*k^5 + 1376459/1069200*k^4 + 25613/89100*k^3 + 64661/77220*k^2 - 31/6435*k, 7909306972/868725*k^14 + 3954653486/173745*k^13 + 282475249/13365*k^12 + 40353607/5346*k^11 - 5764801/8100*k^10 - 823543/648*k^9 - 4857223/19440*k^8 + 487403/19440*k^7 + 472997/97200*k^6 - 20237/19440*k^5 + 32389/35640*k^4 + 1477/5940*k^3 + 5997/7150*k^2 - 2/429*k, 7909306972/868725*k^14 + 27682574402/868725*k^13 + 282475249/6075*k^12 + 4802079233/133650*k^11 + 40353607/2700*k^10 + 40353607/16200*k^9 - 43345253/97200*k^8 - 3411821/12150*k^7 - 2792363/48600*k^6 - 679483/48600*k^5 - 1364797/356400*k^4 - 5831/29700*k^3 + 749/780*k^2 + 5/429*k, 7909306972/868725*k^14 + 3954653486/96525*k^13 + 5367029731/66825*k^12 + 40353607/450*k^11 + 1516142663/24300*k^10 + 149061283/5400*k^9 + 721843843/97200*k^8 + 10369919/10800*k^7 - 737107/12150*k^6 - 10976/225*k^5 - 9189019/1069200*k^4 - 33607/118800*k^3 + 151903/154440*k^2 + 1/78*k, 7909306972/868725*k^14 + 3954653486/78975*k^13 + 8191782221/66825*k^12 + 2138741171/12150*k^11 + 443889677/2700*k^10 + 1694027951/16200*k^9 + 4473670453/97200*k^8 + 168893543/12150*k^7 + 133560427/48600*k^6 + 14182021/48600*k^5 - 3107923/356400*k^4 - 24017/2700*k^3 - 13633/25740*k^2 - 1/13*k, 7909306972/868725*k^14 + 3954653486/66825*k^13 + 11581485209/66825*k^12 + 3672178237/12150*k^11 + 2830517291/8100*k^10 + 4571487193/16200*k^9 + 15871034977/97200*k^8 + 6638916263/97200*k^7 + 1997101379/97200*k^6 + 422458351/97200*k^5 + 13685308/22275*k^4 + 1541491/29700*k^3 + 18097/6435*k^2, 7909306972/868725*k^14 - 3954653486/66825*k^13 + 11581485209/66825*k^12 - 3672178237/12150*k^11 + 2830517291/8100*k^10 - 4571487193/16200*k^9 + 15871034977/97200*k^8 - 830148151/12150*k^7 + 1001954107/48600*k^6 - 215280863/48600*k^5 + 236792353/356400*k^4 - 2010421/29700*k^3 + 135451/25740*k^2 - k]
? getPolys(15)
%23 = [320361328125/14350336*k^15 - 961083984375/7175168*k^14 + 7119140625/19712*k^13 - 1423828125/2464*k^12 + 4372734375/7168*k^11 - 1601015625/3584*k^10 + 5820046875/25088*k^9 - 1075021875/12544*k^8 + 317829125/14336*k^7 - 27562375/7168*k^6 + 15942825/39424*k^5 - 456535/19712*k^4 + 548467/112112*k^3 - 272787/112112*k^2 + 13/14*k, 320361328125/14350336*k^15 - 1601806640625/14350336*k^14 + 505458984375/2050048*k^13 - 7119140625/22528*k^12 + 3714609375/14336*k^11 - 2018671875/14336*k^10 + 5056453125/100352*k^9 - 1113890625/100352*k^8 + 1006625/896*k^7 + 21625/256*k^6 - 1037525/29568*k^5 - 92375/29568*k^4 + 2031787/336336*k^3 - 438331/168168*k^2 + 85/91*k, 320361328125/14350336*k^15 - 320361328125/3587584*k^14 + 7119140625/46592*k^13 - 1423828125/9856*k^12 + 82265625/1024*k^11 - 6328125/256*k^10 + 61453125/25088*k^9 + 5878125/6272*k^8 - 710125/2048*k^7 + 113875/3584*k^6 + 61025/8448*k^5 - 9215/2112*k^4 + 471013/244608*k^3 - 711587/672672*k^2 + 279/364*k, 320361328125/14350336*k^15 - 961083984375/14350336*k^14 + 163740234375/2050048*k^13 - 7119140625/157696*k^12 + 196171875/22528*k^11 + 6328125/2048*k^10 - 188296875/100352*k^9 + 21515625/100352*k^8 + 54125/896*k^7 - 29375/1792*k^6 + 5175/1408*k^5 - 4525/1408*k^4 + 210493/112112*k^3 - 59529/56056*k^2 + 1535/2002*k, 320361328125/14350336*k^15 - 320361328125/7175168*k^14 + 7119140625/256256*k^13 - 499921875/78848*k^11 + 6328125/3584*k^10 + 8296875/25088*k^9 - 2390625/12544*k^8 + 9875/2048*k^7 + 7125/1024*k^6 + 337075/118272*k^5 - 210575/59136*k^4 + 640579/336336*k^3 + 4255/30576*k^2 + 367/1001*k, 320361328125/14350336*k^15 - 320361328125/14350336*k^14 - 7119140625/2050048*k^13 + 1423828125/157696*k^12 - 145546875/157696*k^11 - 18984375/14336*k^10 + 24328125/100352*k^9 + 8746875/100352*k^8 - 17125/896*k^7 - 4625/1792*k^6 + 120775/29568*k^5 + 295/9856*k^4 + 167561/336336*k^3 - 5/56056*k^2 + 1268/3003*k, 320361328125/14350336*k^15 - 7119140625/512512*k^13 + 259453125/78848*k^11 - 9421875/25088*k^9 + 309125/14336*k^7 + 56775/19712*k^5 + 459357/896896*k^3 + 483/1144*k, 320361328125/14350336*k^15 + 320361328125/14350336*k^14 - 7119140625/2050048*k^13 - 1423828125/157696*k^12 - 145546875/157696*k^11 + 18984375/14336*k^10 + 24328125/100352*k^9 - 8746875/100352*k^8 - 17125/896*k^7 + 4625/1792*k^6 + 120775/29568*k^5 - 295/9856*k^4 + 167561/336336*k^3 + 5/56056*k^2 + 1268/3003*k, 320361328125/14350336*k^15 + 320361328125/7175168*k^14 + 7119140625/256256*k^13 - 499921875/78848*k^11 - 6328125/3584*k^10 + 8296875/25088*k^9 + 2390625/12544*k^8 + 9875/2048*k^7 - 7125/1024*k^6 + 337075/118272*k^5 + 210575/59136*k^4 + 640579/336336*k^3 - 4255/30576*k^2 + 367/1001*k, 320361328125/14350336*k^15 + 961083984375/14350336*k^14 + 163740234375/2050048*k^13 + 7119140625/157696*k^12 + 196171875/22528*k^11 - 6328125/2048*k^10 - 188296875/100352*k^9 - 21515625/100352*k^8 + 54125/896*k^7 + 29375/1792*k^6 + 5175/1408*k^5 + 4525/1408*k^4 + 210493/112112*k^3 + 59529/56056*k^2 + 1535/2002*k, 320361328125/14350336*k^15 + 320361328125/3587584*k^14 + 7119140625/46592*k^13 + 1423828125/9856*k^12 + 82265625/1024*k^11 + 6328125/256*k^10 + 61453125/25088*k^9 - 5878125/6272*k^8 - 710125/2048*k^7 - 113875/3584*k^6 + 61025/8448*k^5 + 9215/2112*k^4 + 471013/244608*k^3 + 711587/672672*k^2 + 279/364*k, 320361328125/14350336*k^15 + 1601806640625/14350336*k^14 + 505458984375/2050048*k^13 + 7119140625/22528*k^12 + 3714609375/14336*k^11 + 2018671875/14336*k^10 + 5056453125/100352*k^9 + 1113890625/100352*k^8 + 1006625/896*k^7 - 21625/256*k^6 - 1037525/29568*k^5 + 92375/29568*k^4 + 2031787/336336*k^3 + 438331/168168*k^2 + 85/91*k, 320361328125/14350336*k^15 + 961083984375/7175168*k^14 + 7119140625/19712*k^13 + 1423828125/2464*k^12 + 4372734375/7168*k^11 + 1601015625/3584*k^10 + 5820046875/25088*k^9 + 1075021875/12544*k^8 + 317829125/14336*k^7 + 27562375/7168*k^6 + 15942825/39424*k^5 + 456535/19712*k^4 + 548467/112112*k^3 + 272787/112112*k^2 + 13/14*k, 320361328125/14350336*k^15 + 320361328125/2050048*k^14 + 7119140625/14336*k^13 + 21357421875/22528*k^12 + 17383359375/14336*k^11 + 2246484375/2048*k^10 + 72458578125/100352*k^9 + 5022703125/14336*k^8 + 112460375/896*k^7 + 8423625/256*k^6 + 182934175/29568*k^5 + 3419675/4224*k^4 + 2248439/30576*k^3 + 140311/24024*k^2 + k, 320361328125/14350336*k^15 - 320361328125/2050048*k^14 + 7119140625/14336*k^13 - 21357421875/22528*k^12 + 17383359375/14336*k^11 - 2246484375/2048*k^10 + 72458578125/100352*k^9 - 5022703125/14336*k^8 + 112460375/896*k^7 - 8423625/256*k^6 + 182934175/29568*k^5 - 3419675/4224*k^4 + 2248439/30576*k^3 - 140311/24024*k^2 + k]
? getPolys(16)
%25 = [35184372088832/638512875*k^16 - 17592186044416/49116375*k^15 + 549755813888/521235*k^14 - 2611340115968/1403325*k^13 + 15377056661504/7016625*k^12 - 1158567428096/637875*k^11 + 971140759552/893025*k^10 - 424021065728/893025*k^9 + 95995314176/637875*k^8 - 21670518784/637875*k^7 + 7431669248/1403325*k^6 - 804621568/1403325*k^5 + 12813052376/212837625*k^4 - 60535036/5457375*k^3 + 1731749/675675*k^2 - 14/15*k, 35184372088832/638512875*k^16 - 17592186044416/58046625*k^15 + 95107755802624/127702575*k^14 - 137438953472/127575*k^13 + 7130719453184/7016625*k^12 - 416611827712/637875*k^11 + 258041970688/893025*k^10 - 76841746432/893025*k^9 + 70771392512/4465125*k^8 - 824680448/637875*k^7 - 23046656/280665*k^6 + 2137088/127575*k^5 + 678471736/212837625*k^4 - 13616836/6449625*k^3 + 6286261/4729725*k^2 - 13/15*k, 35184372088832/638512875*k^16 - 17592186044416/70945875*k^15 + 62122406969344/127702575*k^14 - 1099511627776/2027025*k^13 + 2621003792384/7016625*k^12 - 536870912/3375*k^11 + 6652166144/178605*k^10 - 159383552/99225*k^9 - 7095205888/4465125*k^8 + 25649152/70875*k^7 + 48455168/1403325*k^6 - 246272/7425*k^5 + 555091912/70945875*k^4 - 5607892/2627625*k^3 + 2065793/1576575*k^2 - 394/455*k, 35184372088832/638512875*k^16 - 17592186044416/91216125*k^15 + 549755813888/1964655*k^14 - 549755813888/2606175*k^13 + 559419490304/7016625*k^12 - 536870912/91125*k^11 - 6199181312/893025*k^10 + 318767104/127575*k^9 - 520634368/4465125*k^8 - 78340096/637875*k^7 + 46157312/1403325*k^6 + 26624/200475*k^5 - 78371624/212837625*k^4 - 4604716/10135125*k^3 + 158629/363825*k^2 - 971/1365*k, 35184372088832/638512875*k^16 - 17592186044416/127702575*k^15 + 15942918602752/127702575*k^14 - 137438953472/3648645*k^13 - 84825604096/7016625*k^12 + 2147483648/200475*k^11 - 1266679808/893025*k^10 - 23068672/35721*k^9 + 983416832/4465125*k^8 - 4317184/127575*k^7 + 14392832/1403325*k^6 + 15872/40095*k^5 - 11614024/212837625*k^4 - 6719764/14189175*k^3 + 411583/945945*k^2 - 712/1001*k, 35184372088832/638512875*k^16 - 17592186044416/212837625*k^15 + 549755813888/25540515*k^14 + 137438953472/6081075*k^13 - 84825604096/7016625*k^12 - 1073741824/779625*k^11 + 310378496/178605*k^10 - 27262976/297675*k^9 - 353517568/4465125*k^8 - 155648/212625*k^7 - 162304/56133*k^6 + 379648/155925*k^5 + 114622792/70945875*k^4 - 2148364/2627625*k^3 + 130279/315315*k^2 - 10609/15015*k, 35184372088832/638512875*k^16 - 17592186044416/638512875*k^15 - 549755813888/18243225*k^14 + 274877906944/18243225*k^13 + 44023414784/7016625*k^12 - 22011707392/7016625*k^11 - 562036736/893025*k^10 + 281018368/893025*k^9 + 36847616/637875*k^8 - 18423808/637875*k^7 - 9058816/1403325*k^6 + 4529408/1403325*k^5 + 117199912/70945875*k^4 - 58599956/70945875*k^3 + 18607/45045*k^2 - 31826/45045*k, 35184372088832/638512875*k^16 + 17592186044416/638512875*k^15 - 549755813888/18243225*k^14 - 274877906944/18243225*k^13 + 44023414784/7016625*k^12 + 22011707392/7016625*k^11 - 562036736/893025*k^10 - 281018368/893025*k^9 + 36847616/637875*k^8 + 18423808/637875*k^7 - 9058816/1403325*k^6 - 4529408/1403325*k^5 + 117199912/70945875*k^4 + 58599956/70945875*k^3 + 18607/45045*k^2 - 13219/45045*k, 35184372088832/638512875*k^16 + 17592186044416/212837625*k^15 + 549755813888/25540515*k^14 - 137438953472/6081075*k^13 - 84825604096/7016625*k^12 + 1073741824/779625*k^11 + 310378496/178605*k^10 + 27262976/297675*k^9 - 353517568/4465125*k^8 + 155648/212625*k^7 - 162304/56133*k^6 - 379648/155925*k^5 + 114622792/70945875*k^4 + 2148364/2627625*k^3 + 130279/315315*k^2 - 4406/15015*k, 35184372088832/638512875*k^16 + 17592186044416/127702575*k^15 + 15942918602752/127702575*k^14 + 137438953472/3648645*k^13 - 84825604096/7016625*k^12 - 2147483648/200475*k^11 - 1266679808/893025*k^10 + 23068672/35721*k^9 + 983416832/4465125*k^8 + 4317184/127575*k^7 + 14392832/1403325*k^6 - 15872/40095*k^5 - 11614024/212837625*k^4 + 6719764/14189175*k^3 + 411583/945945*k^2 - 289/1001*k, 35184372088832/638512875*k^16 + 17592186044416/91216125*k^15 + 549755813888/1964655*k^14 + 549755813888/2606175*k^13 + 559419490304/7016625*k^12 + 536870912/91125*k^11 - 6199181312/893025*k^10 - 318767104/127575*k^9 - 520634368/4465125*k^8 + 78340096/637875*k^7 + 46157312/1403325*k^6 - 26624/200475*k^5 - 78371624/212837625*k^4 + 4604716/10135125*k^3 + 158629/363825*k^2 - 394/1365*k, 35184372088832/638512875*k^16 + 17592186044416/70945875*k^15 + 62122406969344/127702575*k^14 + 1099511627776/2027025*k^13 + 2621003792384/7016625*k^12 + 536870912/3375*k^11 + 6652166144/178605*k^10 + 159383552/99225*k^9 - 7095205888/4465125*k^8 - 25649152/70875*k^7 + 48455168/1403325*k^6 + 246272/7425*k^5 + 555091912/70945875*k^4 + 5607892/2627625*k^3 + 2065793/1576575*k^2 - 61/455*k, 35184372088832/638512875*k^16 + 17592186044416/58046625*k^15 + 95107755802624/127702575*k^14 + 137438953472/127575*k^13 + 7130719453184/7016625*k^12 + 416611827712/637875*k^11 + 258041970688/893025*k^10 + 76841746432/893025*k^9 + 70771392512/4465125*k^8 + 824680448/637875*k^7 - 23046656/280665*k^6 - 2137088/127575*k^5 + 678471736/212837625*k^4 + 13616836/6449625*k^3 + 6286261/4729725*k^2 - 2/15*k, 35184372088832/638512875*k^16 + 17592186044416/49116375*k^15 + 549755813888/521235*k^14 + 2611340115968/1403325*k^13 + 15377056661504/7016625*k^12 + 1158567428096/637875*k^11 + 971140759552/893025*k^10 + 424021065728/893025*k^9 + 95995314176/637875*k^8 + 21670518784/637875*k^7 + 7431669248/1403325*k^6 + 804621568/1403325*k^5 + 12813052376/212837625*k^4 + 60535036/5457375*k^3 + 1731749/675675*k^2 - 1/15*k, 35184372088832/638512875*k^16 + 17592186044416/42567525*k^15 + 25838523252736/18243225*k^14 + 274877906944/93555*k^13 + 28906203643904/7016625*k^12 + 58518929408/14175*k^11 + 2722178793472/893025*k^10 + 20061356032/11907*k^9 + 447107710976/637875*k^8 + 9306251264/42525*k^7 + 71193523712/1403325*k^6 + 267506432/31185*k^5 + 73379741992/70945875*k^4 + 138523948/1576575*k^3 + 271681/45045*k^2, 35184372088832/638512875*k^16 - 17592186044416/42567525*k^15 + 25838523252736/18243225*k^14 - 274877906944/93555*k^13 + 28906203643904/7016625*k^12 - 58518929408/14175*k^11 + 2722178793472/893025*k^10 - 20061356032/11907*k^9 + 447107710976/637875*k^8 - 9306251264/42525*k^7 + 71193523712/1403325*k^6 - 267506432/31185*k^5 + 73379741992/70945875*k^4 - 138523948/1576575*k^3 + 271681/45045*k^2 - k]

但是这个表达式很难看，考虑到结果应该很接近$\frac{C_n^m}m$,我们从结果中减去$\frac{C_n^m}m$,得到
? getPolysd(2)
%27 = [-1/2*k, -1/2*k]
? getPolysd(3)
%28 = [2/3*k, 2/3*k, 2/3*k]
? getPolysd(4)
%29 = [1/2*k^2 - 3/4*k, 1/2*k^2 - 1/4*k, 1/2*k^2 - 1/4*k, 1/2*k^2 - 3/4*k]
? getPolysd(5)
%30 = [4/5*k, 4/5*k, 4/5*k, 4/5*k, 4/5*k]
? getPolysd(6)
%31 = [-3/4*k^3 + 17/12*k^2 - 5/6*k, -3/4*k^3 + 2/3*k^2 - 7/12*k, -3/4*k^3 + 2/3*k^2 + 1/12*k, -3/4*k^3 - 1/12*k^2 - 1/6*k, -3/4*k^3 - 1/12*k^2 - 1/6*k, -3/4*k^3 + 17/12*k^2 - 5/6*k]
? getPolysd(7)
%32 = [6/7*k, 6/7*k, 6/7*k, 6/7*k, 6/7*k, 6/7*k, 6/7*k]
? getPolysd(8)
%33 = [4/3*k^4 - 2*k^3 + 17/12*k^2 - 7/8*k, 4/3*k^4 - 2/3*k^3 + 5/12*k^2 - 17/24*k, 4/3*k^4 - 2/3*k^3 + 5/12*k^2 - 17/24*k, 4/3*k^4 + 2/3*k^3 + 5/12*k^2 - 7/24*k, 4/3*k^4 + 2/3*k^3 + 5/12*k^2 - 7/24*k, 4/3*k^4 + 2*k^3 + 17/12*k^2 - 1/8*k, 4/3*k^4 + 2*k^3 + 17/12*k^2 - 1/8*k, 4/3*k^4 - 2*k^3 + 17/12*k^2 - 7/8*k]
? getPolysd(9)
%34 = [k^3 - k^2 + 8/9*k, k^3 - k^2 + 8/9*k, k^3 + 5/9*k, k^3 + 5/9*k, k^3 + 5/9*k, k^3 + k^2 + 8/9*k, k^3 + k^2 + 8/9*k, k^3 + k^2 + 8/9*k, k^3 - k^2 + 8/9*k]
? getPolysd(10)
%35 = [-125/48*k^5 + 125/24*k^4 - 175/48*k^3 + 221/120*k^2 - 9/10*k, -125/48*k^5 + 125/48*k^4 - 25/48*k^3 + 167/240*k^2 - 31/40*k, -125/48*k^5 + 125/48*k^4 - 25/48*k^3 + 167/240*k^2 - 31/40*k, -125/48*k^5 + 25/48*k^3 + 4/5*k^2 - 49/60*k, -125/48*k^5 + 25/48*k^3 + 4/5*k^2 - 1/60*k, -125/48*k^5 - 125/48*k^4 - 25/48*k^3 + 217/240*k^2 + 1/40*k, -125/48*k^5 - 125/48*k^4 - 25/48*k^3 + 217/240*k^2 + 1/40*k, -125/48*k^5 - 125/24*k^4 - 175/48*k^3 - 29/120*k^2 - 1/10*k, -125/48*k^5 - 125/24*k^4 - 175/48*k^3 - 29/120*k^2 - 1/10*k, -125/48*k^5 + 125/24*k^4 - 175/48*k^3 + 221/120*k^2 - 9/10*k]
? getPolysd(11)
%36 = [10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k, 10/11*k]
? getPolysd(12)
%37 = [27/5*k^6 - 27/2*k^5 + 523/36*k^4 - 217/24*k^3 + 1241/360*k^2 - 11/12*k, 27/5*k^6 - 81/10*k^5 + 199/36*k^4 - 91/24*k^3 + 791/360*k^2 - 49/60*k, 27/5*k^6 - 81/10*k^5 + 199/36*k^4 - 145/72*k^3 + 311/360*k^2 - 107/180*k, 27/5*k^6 - 27/10*k^5 + 37/36*k^4 - 91/72*k^3 + 43/180*k^2 - 133/360*k, 27/5*k^6 - 27/10*k^5 + 37/36*k^4 - 91/72*k^3 + 43/180*k^2 - 133/360*k, 27/5*k^6 + 27/10*k^5 + 37/36*k^4 - 17/72*k^3 + 43/180*k^2 - 47/360*k, 27/5*k^6 + 27/10*k^5 + 37/36*k^4 - 17/72*k^3 + 43/180*k^2 - 47/360*k, 27/5*k^6 + 81/10*k^5 + 199/36*k^4 + 37/72*k^3 - 229/360*k^2 - 73/180*k, 27/5*k^6 + 81/10*k^5 + 199/36*k^4 + 55/24*k^3 + 251/360*k^2 - 11/60*k, 27/5*k^6 + 27/2*k^5 + 523/36*k^4 + 181/24*k^3 + 701/360*k^2 - 1/12*k, 27/5*k^6 + 27/2*k^5 + 523/36*k^4 + 181/24*k^3 + 701/360*k^2 - 1/12*k, 27/5*k^6 - 27/2*k^5 + 523/36*k^4 - 217/24*k^3 + 1241/360*k^2 - 11/12*k]
? getPolysd(13)
%38 = [12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k, 12/13*k]
? getPolysd(14)
%39 = [-16807/1440*k^7 + 16807/480*k^6 - 12005/288*k^5 + 2401/96*k^4 - 1421/180*k^3 + 583/280*k^2 - 13/14*k, -16807/1440*k^7 + 16807/720*k^6 - 2401/144*k^5 + 343/72*k^4 - 343/1440*k^3 + 3781/5040*k^2 - 71/84*k, -16807/1440*k^7 + 16807/720*k^6 - 2401/144*k^5 + 343/72*k^4 - 343/1440*k^3 + 3781/5040*k^2 - 71/84*k, -16807/1440*k^7 + 16807/1440*k^6 - 2401/1440*k^5 - 343/288*k^4 + 49/180*k^3 + 2209/2520*k^2 - 181/210*k, -16807/1440*k^7 + 16807/1440*k^6 - 2401/1440*k^5 - 343/288*k^4 + 49/180*k^3 + 2209/2520*k^2 - 181/210*k, -16807/1440*k^7 + 2401/720*k^5 - 343/1440*k^3 + 6/7*k^2 - 239/280*k, -16807/1440*k^7 + 2401/720*k^5 - 343/1440*k^3 + 6/7*k^2 + 1/280*k, -16807/1440*k^7 - 16807/1440*k^6 - 2401/1440*k^5 + 343/288*k^4 + 49/180*k^3 + 2111/2520*k^2 - 1/210*k, -16807/1440*k^7 - 16807/1440*k^6 - 2401/1440*k^5 + 343/288*k^4 + 49/180*k^3 + 2111/2520*k^2 - 1/210*k, -16807/1440*k^7 - 16807/720*k^6 - 2401/144*k^5 - 343/72*k^4 - 343/1440*k^3 + 4859/5040*k^2 + 1/84*k, -16807/1440*k^7 - 16807/720*k^6 - 2401/144*k^5 - 343/72*k^4 - 343/1440*k^3 + 4859/5040*k^2 + 1/84*k, -16807/1440*k^7 - 16807/480*k^6 - 12005/288*k^5 - 2401/96*k^4 - 1421/180*k^3 - 103/280*k^2 - 1/14*k, -16807/1440*k^7 - 16807/480*k^6 - 12005/288*k^5 - 2401/96*k^4 - 1421/180*k^3 - 103/280*k^2 - 1/14*k, -16807/1440*k^7 + 16807/480*k^6 - 12005/288*k^5 + 2401/96*k^4 - 1421/180*k^3 + 583/280*k^2 - 13/14*k]
? getPolysd(15)
%40 = [125/36*k^5 - 125/18*k^4 + 1091/180*k^3 - 233/90*k^2 + 14/15*k, 125/36*k^5 - 125/18*k^4 + 1091/180*k^3 - 233/90*k^2 + 14/15*k, 125/36*k^5 - 125/36*k^4 + 341/180*k^3 - 191/180*k^2 + 23/30*k, 125/36*k^5 - 125/36*k^4 + 341/180*k^3 - 191/180*k^2 + 23/30*k, 125/36*k^5 - 125/36*k^4 + 341/180*k^3 + 5/36*k^2 + 11/30*k, 125/36*k^5 + 91/180*k^3 + 19/45*k, 125/36*k^5 + 91/180*k^3 + 19/45*k, 125/36*k^5 + 91/180*k^3 + 19/45*k, 125/36*k^5 + 125/36*k^4 + 341/180*k^3 - 5/36*k^2 + 11/30*k, 125/36*k^5 + 125/36*k^4 + 341/180*k^3 + 191/180*k^2 + 23/30*k, 125/36*k^5 + 125/36*k^4 + 341/180*k^3 + 191/180*k^2 + 23/30*k, 125/36*k^5 + 125/18*k^4 + 1091/180*k^3 + 233/90*k^2 + 14/15*k, 125/36*k^5 + 125/18*k^4 + 1091/180*k^3 + 233/90*k^2 + 14/15*k, 125/36*k^5 + 125/18*k^4 + 1091/180*k^3 + 233/90*k^2 + 14/15*k, 125/36*k^5 - 125/18*k^4 + 1091/180*k^3 - 233/90*k^2 + 14/15*k]
? getPolysd(16)
%41 = [8192/315*k^8 - 4096/45*k^7 + 5888/45*k^6 - 896/9*k^5 + 1994/45*k^4 - 559/45*k^3 + 2279/840*k^2 - 15/16*k, 8192/315*k^8 - 4096/63*k^7 + 2816/45*k^6 - 256/9*k^5 + 314/45*k^4 - 19/9*k^3 + 1103/840*k^2 - 97/112*k, 8192/315*k^8 - 4096/63*k^7 + 2816/45*k^6 - 256/9*k^5 + 314/45*k^4 - 19/9*k^3 + 1103/840*k^2 - 97/112*k, 8192/315*k^8 - 4096/105*k^7 + 256/15*k^6 - 2/15*k^4 - 7/15*k^3 + 1097/2520*k^2 - 239/336*k, 8192/315*k^8 - 4096/105*k^7 + 256/15*k^6 - 2/15*k^4 - 7/15*k^3 + 1097/2520*k^2 - 239/336*k, 8192/315*k^8 - 4096/315*k^7 - 256/45*k^6 + 128/45*k^5 + 74/45*k^4 - 37/45*k^3 + 347/840*k^2 - 1187/1680*k, 8192/315*k^8 - 4096/315*k^7 - 256/45*k^6 + 128/45*k^5 + 74/45*k^4 - 37/45*k^3 + 347/840*k^2 - 1187/1680*k, 8192/315*k^8 + 4096/315*k^7 - 256/45*k^6 - 128/45*k^5 + 74/45*k^4 + 37/45*k^3 + 347/840*k^2 - 493/1680*k, 8192/315*k^8 + 4096/315*k^7 - 256/45*k^6 - 128/45*k^5 + 74/45*k^4 + 37/45*k^3 + 347/840*k^2 - 493/1680*k, 8192/315*k^8 + 4096/105*k^7 + 256/15*k^6 - 2/15*k^4 + 7/15*k^3 + 1097/2520*k^2 - 97/336*k, 8192/315*k^8 + 4096/105*k^7 + 256/15*k^6 - 2/15*k^4 + 7/15*k^3 + 1097/2520*k^2 - 97/336*k, 8192/315*k^8 + 4096/63*k^7 + 2816/45*k^6 + 256/9*k^5 + 314/45*k^4 + 19/9*k^3 + 1103/840*k^2 - 15/112*k, 8192/315*k^8 + 4096/63*k^7 + 2816/45*k^6 + 256/9*k^5 + 314/45*k^4 + 19/9*k^3 + 1103/840*k^2 - 15/112*k, 8192/315*k^8 + 4096/45*k^7 + 5888/45*k^6 + 896/9*k^5 + 1994/45*k^4 + 559/45*k^3 + 2279/840*k^2 - 1/16*k, 8192/315*k^8 + 4096/45*k^7 + 5888/45*k^6 + 896/9*k^5 + 1994/45*k^4 + 559/45*k^3 + 2279/840*k^2 - 1/16*k, 8192/315*k^8 - 4096/45*k^7 + 5888/45*k^6 - 896/9*k^5 + 1994/45*k^4 - 559/45*k^3 + 2279/840*k^2 - 15/16*k]
? getPolysd(18)
%42 = [-531441/8960*k^9 + 531441/2240*k^8 - 255879/640*k^7 + 60201/160*k^6 - 282807/1280*k^5 + 27067/320*k^4 - 47451/2240*k^3 + 20401/5040*k^2 - 17/18*k, -531441/8960*k^9 + 1594323/8960*k^8 - 137781/640*k^7 + 89901/640*k^6 - 78687/1280*k^5 + 29941/1280*k^4 - 1119/140*k^3 + 52201/20160*k^2 - 127/144*k, -531441/8960*k^9 + 1594323/8960*k^8 - 137781/640*k^7 + 89901/640*k^6 - 69471/1280*k^5 + 14581/1280*k^4 - 139/140*k^3 + 18601/20160*k^2 - 539/720*k, -531441/8960*k^9 + 531441/4480*k^8 - 373977/4480*k^7 + 1773/64*k^6 - 11151/1280*k^5 + 2147/640*k^4 - 1909/2240*k^3 + 2095/2016*k^2 - 1909/2520*k, -531441/8960*k^9 + 531441/4480*k^8 - 373977/4480*k^7 + 1773/64*k^6 - 11151/1280*k^5 + 2147/640*k^4 - 1909/2240*k^3 + 2095/2016*k^2 - 1909/2520*k, -531441/8960*k^9 + 531441/8960*k^8 - 19683/4480*k^7 - 1953/640*k^6 - 387/256*k^5 - 713/1280*k^4 - 59/560*k^3 + 13807/20160*k^2 - 3131/5040*k, -531441/8960*k^9 + 531441/8960*k^8 - 19683/4480*k^7 - 1953/640*k^6 - 387/256*k^5 - 713/1280*k^4 - 59/560*k^3 + 13807/20160*k^2 - 3131/5040*k, -531441/8960*k^9 + 19683/896*k^7 + 36/5*k^6 - 7767/1280*k^5 - k^4 + 41/448*k^3 + 31/45*k^2 - 157/252*k, -531441/8960*k^9 + 19683/896*k^7 + 36/5*k^6 + 1449/1280*k^5 - k^4 - 407/448*k^3 + 31/45*k^2 + 83/1260*k, -531441/8960*k^9 - 531441/8960*k^8 - 19683/4480*k^7 + 11169/640*k^6 + 7281/1280*k^5 - 1847/1280*k^4 - 619/560*k^3 + 13969/20160*k^2 + 341/5040*k, -531441/8960*k^9 - 531441/8960*k^8 - 19683/4480*k^7 + 11169/640*k^6 + 7281/1280*k^5 - 1847/1280*k^4 - 619/560*k^3 + 13969/20160*k^2 + 341/5040*k, -531441/8960*k^9 - 531441/4480*k^8 - 373977/4480*k^7 - 4257/320*k^6 + 16497/1280*k^5 + 4253/640*k^4 + 331/2240*k^3 + 53/10080*k^2 - 341/2520*k, -531441/8960*k^9 - 531441/4480*k^8 - 373977/4480*k^7 - 4257/320*k^6 + 16497/1280*k^5 + 4253/640*k^4 + 331/2240*k^3 + 53/10080*k^2 - 341/2520*k, -531441/8960*k^9 - 1594323/8960*k^8 - 137781/640*k^7 - 16137/128*k^6 - 41823/1280*k^5 - 1781/1280*k^4 + 1/140*k^3 + 491/4032*k^2 - 91/720*k, -531441/8960*k^9 - 1594323/8960*k^8 - 137781/640*k^7 - 16137/128*k^6 - 32607/1280*k^5 + 13579/1280*k^4 + 981/140*k^3 + 7211/4032*k^2 + 1/144*k, -531441/8960*k^9 - 531441/2240*k^8 - 255879/640*k^7 - 57897/160*k^6 - 236727/1280*k^5 - 16187/320*k^4 - 13851/2240*k^3 + 1663/5040*k^2 - 1/18*k, -531441/8960*k^9 - 531441/2240*k^8 - 255879/640*k^7 - 57897/160*k^6 - 236727/1280*k^5 - 16187/320*k^4 - 13851/2240*k^3 + 1663/5040*k^2 - 1/18*k, -531441/8960*k^9 + 531441/2240*k^8 - 255879/640*k^7 + 60201/160*k^6 - 282807/1280*k^5 + 27067/320*k^4 - 47451/2240*k^3 + 20401/5040*k^2 - 17/18*k]
? getPolysd(20)
%43 = [78125/567*k^10 - 78125/126*k^9 + 453125/378*k^8 - 15625/12*k^7 + 376625/432*k^6 - 35875/96*k^5 + 2428009/22680*k^4 - 231881/10080*k^3 + 21787/5040*k^2 - 19/20*k, 78125/567*k^10 - 78125/162*k^9 + 265625/378*k^8 - 59375/108*k^7 + 106625/432*k^6 - 55625/864*k^5 + 40673/2835*k^4 - 87197/12960*k^3 + 14177/5040*k^2 - 161/180*k, 78125/567*k^10 - 78125/162*k^9 + 265625/378*k^8 - 59375/108*k^7 + 106625/432*k^6 - 55625/864*k^5 + 40673/2835*k^4 - 87197/12960*k^3 + 14177/5040*k^2 - 161/180*k, 78125/567*k^10 - 390625/1134*k^9 + 62500/189*k^8 - 15625/108*k^7 + 9125/432*k^6 + 1625/864*k^5 + 269911/90720*k^4 - 327679/90720*k^3 + 1991/1120*k^2 - 559/720*k, 78125/567*k^10 - 390625/1134*k^9 + 62500/189*k^8 - 15625/108*k^7 + 9125/432*k^6 + 1625/864*k^5 + 269911/90720*k^4 - 134143/90720*k^3 + 199/1120*k^2 - 367/720*k, 78125/567*k^10 - 78125/378*k^9 + 15625/189*k^8 + 3125/252*k^7 - 5875/432*k^6 - 125/96*k^5 + 474661/90720*k^4 - 50231/30240*k^3 + 1601/10080*k^2 - 853/1680*k, 78125/567*k^10 - 78125/378*k^9 + 15625/189*k^8 + 3125/252*k^7 - 5875/432*k^6 - 125/96*k^5 + 474661/90720*k^4 - 50231/30240*k^3 + 1601/10080*k^2 - 853/1680*k, 78125/567*k^10 - 78125/1134*k^9 - 15625/378*k^8 + 15625/756*k^7 + 1625/432*k^6 - 3875/864*k^5 + 91643/45360*k^4 - 44393/90720*k^3 + 337/1260*k^2 - 1387/2520*k, 78125/567*k^10 - 78125/1134*k^9 - 15625/378*k^8 + 15625/756*k^7 + 1625/432*k^6 - 3875/864*k^5 + 91643/45360*k^4 - 44393/90720*k^3 + 337/1260*k^2 - 1387/2520*k, 78125/567*k^10 + 78125/1134*k^9 - 15625/378*k^8 - 15625/756*k^7 + 1625/432*k^6 - 625/864*k^5 + 91643/45360*k^4 + 138893/90720*k^3 + 337/1260*k^2 - 713/2520*k, 78125/567*k^10 + 78125/1134*k^9 - 15625/378*k^8 - 15625/756*k^7 + 1625/432*k^6 - 625/864*k^5 + 91643/45360*k^4 + 138893/90720*k^3 + 337/1260*k^2 - 713/2520*k, 78125/567*k^10 + 78125/378*k^9 + 15625/189*k^8 - 3125/252*k^7 - 5875/432*k^6 - 125/32*k^5 + 2161/90720*k^4 + 18731/30240*k^3 + 3701/10080*k^2 - 407/1680*k, 78125/567*k^10 + 78125/378*k^9 + 15625/189*k^8 - 3125/252*k^7 - 5875/432*k^6 - 125/32*k^5 + 2161/90720*k^4 + 18731/30240*k^3 + 3701/10080*k^2 - 407/1680*k, 78125/567*k^10 + 390625/1134*k^9 + 62500/189*k^8 + 15625/108*k^7 + 9125/432*k^6 - 6125/864*k^5 - 202589/90720*k^4 + 39643/90720*k^3 + 1297/3360*k^2 - 173/720*k, 78125/567*k^10 + 390625/1134*k^9 + 62500/189*k^8 + 15625/108*k^7 + 9125/432*k^6 - 6125/864*k^5 - 202589/90720*k^4 + 233179/90720*k^3 + 6673/3360*k^2 + 19/720*k, 78125/567*k^10 + 78125/162*k^9 + 265625/378*k^8 + 59375/108*k^7 + 106625/432*k^6 + 51125/864*k^5 + 44567/11340*k^4 - 7303/12960*k^3 + 3677/5040*k^2 - 19/180*k, 78125/567*k^10 + 78125/162*k^9 + 265625/378*k^8 + 59375/108*k^7 + 106625/432*k^6 + 51125/864*k^5 + 44567/11340*k^4 - 7303/12960*k^3 + 3677/5040*k^2 - 19/180*k, 78125/567*k^10 + 78125/126*k^9 + 453125/378*k^8 + 15625/12*k^7 + 376625/432*k^6 + 35375/96*k^5 + 2191759/22680*k^4 + 158381/10080*k^3 + 11287/5040*k^2 - 1/20*k, 78125/567*k^10 + 78125/126*k^9 + 453125/378*k^8 + 15625/12*k^7 + 376625/432*k^6 + 35375/96*k^5 + 2191759/22680*k^4 + 158381/10080*k^3 + 11287/5040*k^2 - 1/20*k, 78125/567*k^10 - 78125/126*k^9 + 453125/378*k^8 - 15625/12*k^7 + 376625/432*k^6 - 35875/96*k^5 + 2428009/22680*k^4 - 231881/10080*k^3 + 21787/5040*k^2 - 19/20*k]
? getPolysd(21)
%44 = [16807/1080*k^7 - 16807/360*k^6 + 12005/216*k^5 - 2401/72*k^4 + 11162/945*k^3 - 613/210*k^2 + 20/21*k, 16807/1080*k^7 - 16807/360*k^6 + 12005/216*k^5 - 2401/72*k^4 + 11162/945*k^3 - 613/210*k^2 + 20/21*k, 16807/1080*k^7 - 16807/540*k^6 + 2401/108*k^5 - 343/54*k^4 + 12121/7560*k^3 - 4321/3780*k^2 + 53/63*k, 16807/1080*k^7 - 16807/540*k^6 + 2401/108*k^5 - 343/54*k^4 + 12121/7560*k^3 - 4321/3780*k^2 + 53/63*k, 16807/1080*k^7 - 16807/540*k^6 + 2401/108*k^5 - 343/54*k^4 + 12121/7560*k^3 - 4321/3780*k^2 + 53/63*k, 16807/1080*k^7 - 16807/1080*k^6 + 2401/1080*k^5 + 343/216*k^4 + 872/945*k^3 - 2479/1890*k^2 + 272/315*k, 16807/1080*k^7 - 16807/1080*k^6 + 2401/1080*k^5 + 343/216*k^4 + 872/945*k^3 - 7/270*k^2 + 137/315*k, 16807/1080*k^7 - 16807/1080*k^6 + 2401/1080*k^5 + 343/216*k^4 + 872/945*k^3 - 7/270*k^2 + 137/315*k, 16807/1080*k^7 - 2401/540*k^5 + 12121/7560*k^3 + 89/210*k, 16807/1080*k^7 - 2401/540*k^5 + 12121/7560*k^3 + 89/210*k, 16807/1080*k^7 - 2401/540*k^5 + 12121/7560*k^3 + 89/210*k, 16807/1080*k^7 + 16807/1080*k^6 + 2401/1080*k^5 - 343/216*k^4 + 872/945*k^3 + 7/270*k^2 + 137/315*k, 16807/1080*k^7 + 16807/1080*k^6 + 2401/1080*k^5 - 343/216*k^4 + 872/945*k^3 + 7/270*k^2 + 137/315*k, 16807/1080*k^7 + 16807/1080*k^6 + 2401/1080*k^5 - 343/216*k^4 + 872/945*k^3 + 2479/1890*k^2 + 272/315*k, 16807/1080*k^7 + 16807/540*k^6 + 2401/108*k^5 + 343/54*k^4 + 12121/7560*k^3 + 4321/3780*k^2 + 53/63*k, 16807/1080*k^7 + 16807/540*k^6 + 2401/108*k^5 + 343/54*k^4 + 12121/7560*k^3 + 4321/3780*k^2 + 53/63*k, 16807/1080*k^7 + 16807/540*k^6 + 2401/108*k^5 + 343/54*k^4 + 12121/7560*k^3 + 4321/3780*k^2 + 53/63*k, 16807/1080*k^7 + 16807/360*k^6 + 12005/216*k^5 + 2401/72*k^4 + 11162/945*k^3 + 613/210*k^2 + 20/21*k, 16807/1080*k^7 + 16807/360*k^6 + 12005/216*k^5 + 2401/72*k^4 + 11162/945*k^3 + 613/210*k^2 + 20/21*k, 16807/1080*k^7 + 16807/360*k^6 + 12005/216*k^5 + 2401/72*k^4 + 11162/945*k^3 + 613/210*k^2 + 20/21*k, 16807/1080*k^7 - 16807/360*k^6 + 12005/216*k^5 - 2401/72*k^4 + 11162/945*k^3 - 613/210*k^2 + 20/21*k]

而从上面结论中可以看出在m为奇素数时，更简洁的表达式为$F(n,m)=\frac{C_n^m}m+\frac{m-1}m\lfloor\frac nm\rfloor$  或者$F(n,m)=\frac{C_n^m-\lfloor\frac nm\rfloor}m+\lfloor\frac nm\rfloor$ 。

王守恩

王守恩 发表于 2021-8-5 11:12
这题目不太好做。

先看一串数。

这题目还是不太好做。

再看一串数(应该是这串数吧？我们已经有6个数了，请网友验证)。

m=02，通项公式1次项(最高项)系数=1/2
m=04，通项公式2次项(最高项)系数=1/2
m=06，通项公式3次项(最高项)系数=3/4
m=08，通项公式4次项(最高项)系数=4/3
m=10，通项公式5次项(最高项)系数=125/48
m=12，通项公式6次项(最高项)系数=27/5
......
{1/2,  1/2,  3/4,  4/3,  125/48,  27/5,  16807/1440,  8192/315,  531441/8960,  78125/567,  2357947691/7257600,
1492992/1925, 1792160394037/958003200, 3954653486/868725, 320361328125/28700672, 17592186044416/638512875,
2862423051509815793/41845579776000,  2541865828329/14889875,  5480386857784802185939/12804747411456000,
16000000000000000/14849255421, 41209797661291758429/15135211520000, 122318180896829092582/17717861581875,
39471584120695485887249589623/2248001455555215360000,   112190507323565801472/2505147019375, ......}

通项公式\(=\frac{m^m}{2m*m!}\)

mathe

公式为\[\sum_{d|m} c(m/d) C_{[dn/m]}^d\]
其中系数c(m/d)需要测试一下，好像c(2)=-1,其它情况c(x)=x-1? 不过还需要检验一下，最后一项系数好像不对，但是大部分对上了

mathe

\[\frac1m\sum_{d|m} c(m,d) C_{[dn/m]}^d\]
c(m,d)的分布
c(3,.)={1->2, 3->1}
c(4,.)={1->-2,2->1,4->1}
c(5,.)={1->4,5->1}
c(6,.)={1->-2,2->2,3->-1,6->1}
c(7,.)={1->6,7->1}
c(8,.)={1->-4,2->2,4->1,8->1}
c(9,.)={1->6,3->2,9->1}
c(10,.)={1->-4,2->4,5->-1,10->1}
c(11,.)={1->10,11->1}
c(12,.)={1->-4,2->2,3->-2,4->2,6->1,12->1}
c(13,.)={1->12,13->1}
c(14,.)={1->-6,2->6,7->-1,14->1}
c(15,.)={1->8,3->4,5->2,15->1}
c(16,.)={1->-8,2->4,4->2,8->1,16->1}
c(18,.)={1->-6,2->6,3->-2,6->2,9->-1,18->1}
c(20,.)={1->-8,2->4,4->4,5->-2,10->1,20->1}
c(21,.)={1->12,3->6,7->2,21->1}
c(22,.)={1->-10,2->10,11->-1,22->1}
c(24,.)={1->-8,2->4,3->-4,4->2,6->2,8->2,12->1,24->1}
c(25,.)={1->20,5->4,25->1}
c(26,.)={1->-12,2->12,13->-1,26->1}
c(27,.)={1->18,3->6,9->2,27->1}
c(28,.)={1->-12,2->6,4->6,7->-2,14->1,28->1}
c(30,.)={1->-8,2->8,3->-4,5->-2,6->4,10->2,15->-1,30->1}
c(32,.)={1->-16,2->8,4->4,8->2,16->1,32->1}
c(33,.)={1->20,3->10,11->2,33->1}
c(34,.)={1->-16,2->16,17->-1,34->1}
c(35,.)={1->24,5->6,7->4,35->1}
c(36,.)={1->-12,2->6,3->-4,4->6,6->2,9->-2,12->2,18->1,36->1}
所以$C(m,d)=(-1)^(m-d) \varphi(\frac md)$, 即\[F(n,m)=\frac1m\sum_{d|m} (-1)^{m-d} \varphi\left(\frac md\right) C_{[dn/m]}^d\]

hujunhua

`C_{[dn/m]}^d`小罐装大罐，吃相有点难看呵 ，咱们换大点的罐子\[
\text{F}(n,m)=\frac1m\sum_{d|m} (-1)^{m-d} \varphi\left(\frac md\right)\begin{pmatrix}[dn/m]\\d\end{pmatrix}
\]

mathe

如果我们继续扩展，好像可以有
\[F_{h}(n,m)=C_M(M,n,h)=\frac1m\sum_{d|m}c(m,d,h){[dn/m]\choose d}\]
其中表格c(m,d,h)我现在还没有找出规律
比如m=12,横向为d,纵向为h,可以有表格
\[\begin{matrix}m=12&\begin{matrix}1&2&3&4&6&12\end{matrix}\\
\begin{matrix} 1\\ 2\\ 3\\ 4\\ 6\\ 12\end{matrix}&\begin{bmatrix}0&1&0&-1&-1&1\\-2&-1&2&-1&1&1\\0&-2&0&2&-1&1\\2&-1&-2&-1&1&1\\4&2&2&2&1&1\\-4&2&-2&2&1&1\end{bmatrix}\end{matrix}\]
而比如m=8,可以有
\[\begin{matrix}m=8&\begin{matrix}1&2&4&8\end{matrix}\\
\begin{matrix} 1\\ 2\\ 4\\ 8\end{matrix}&
\begin{bmatrix}0&0&-1&1\\0&-2&1&1\\4&2&1&1\\-4&2&1&1\end{bmatrix}\end{matrix}\]
而比如m=4,可以有
\[\begin{matrix}m=4&\begin{matrix}1&2&4\end{matrix}\\
\begin{matrix} 1\\ 2\\ 4\end{matrix}&
\begin{bmatrix}0&-1&1\\2&1&1\\-2&1&1\end{bmatrix}\end{matrix}\]
而比如m=9,可以有
\[\begin{matrix}m=9&\begin{matrix}1&3&9\end{matrix}\\
\begin{matrix} 1\\ 3\\ 9\end{matrix}&
\begin{bmatrix}0&-1&1\\-3&2&1\\6&2&1\end{bmatrix}\end{matrix}\]
而比如m=16,可以有
\[\begin{matrix}m=16&\begin{matrix}1&2&4&8&16\end{matrix}\\
\begin{matrix} 1\\ 2\\ 4\\ 8\\ 16\end{matrix}&
\begin{bmatrix}0&0&0&-1&1\\0&0&-2&1&1\\0&-4&2&1&1\\8&4&2&1&1\\-8&4&2&1&1\end{bmatrix}\end{matrix}\]
分析结果好像如下,定义有理数域中函数g(x)如下,
\(g(p^a)=\begin{cases}0&a\ge2\\\frac1{1-p}&a==1\\1&a\le0\end{cases}\)
\(g(p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t})=g(p_1^{a_1})g(p_2^{a_2})\cdots g(p_t^{a_t})\)
那么$c(m,d,h)=(-1)^{m-d}\varphi(\frac m d)g(\frac{m}{dh})$
即
\[F_{h}(n,m)=C_M(M,n,h)=\frac1m\sum_{d|m}(-1)^{m-d}\varphi\left(\frac m d\right)g\left(\frac{m}{dh}\right){[dn/m]\choose d}\]

mathe

1. CNM(n,m)=if(n<m,0,n!/m!/(n-m)!)
   
2. getG(n)={
   
3.     local(v,p);
   
4.     v=factor(n);p=1;
   
5.     for(x=1,length(v[,1]),
   
6.         if(v[x,2]>=2, p=0;break());
   
7.         if(v[x,2]==1, p*=1/(1-v[x,1]))
   
8.     );
   
9.     p
   
10. }
    
11. Fh(n,m,h)={
    
12.     local(s,k,d);
    
13.     s=0;d=divisors(m);
    
14.     for(u=1,length(d),
    
15.         s+=(-1)^(m-d[u])*eulerphi(m/d[u])*getG(m/(d[u]*gcd(h,m)))*CNM(floor(d[u]*n/m),d[u])
    
16.     );
    
17.     s/m
    
18. }

复制代码

上面代码出来的结果我验算了$F_3(1..200,15), F_{15}(1..2000,180)$,结果都能够匹配，所以应该问题不大了

王守恩

mathe 发表于 2021-8-16 13:16
M=12

尊敬的mathe! 太厉害了！不是想学就能学的。只能仰望，差距太大，得用光年计算：一万光年。

看33楼：接下来25应该是最简单的，我还就不信(6个数能找出规律，数多了反而不好找了？)
在 1～n 中任意取 25 个不同整数，使得这 25 个数之和能被 25 整除，有几种不同取法？

Table[\(\D\sum_{k=24}^n\)(Count[Total/@Subsets[Range@k, {24}]/25, _Integer]), {n, 24,33}]

{1, 2, 15, 132, 951, 5702, 29453, 134636, 555368, 2098052}

这里是n=33(再怎么鼓捣，出不来了)，如果要搞个通项，需要n=25——95，能来几项？谢谢！