考虑$1$个长度为$n$，有以下$8$种符号的序列：
、、、、、、、，
其中，“http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png”表示空格。
这个序列需要符合以下要求：
$1$、必须是合法的括号序列：左右括号必须能一一配对。配对时，左括号在前、右括号在后，左圆括号必须和右圆括号配对，左方括号必须和右方括号配对。
$2$、序列里要么有且只有$1$个http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.png并且没有http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.png，要么有且只有$1$个http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.png并且没有http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.png。
$3$、方括号不能放在任何一对圆括号里面。
$4$、不能出现以下子序列：
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/130423002800f5e53b5c481cf9.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/130423002800f5e53b5c481cf9.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028828cc84237530d5c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028498085635d27d67e.png
问：长度为$n$的，符合要求的序列有多少个？
例如，长度为$5$的符合要求的序列有$33$个：
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028ebe748951da87230.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.png
http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
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http://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/13042300283f7b0fdac22f4db6.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028414ee67defc2f1e0.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028580574ecfb79292a.pnghttp://gg.a.5d6d.com/userdirs/7/4/emath/attachments/month_1304/1304230028a047d2d7baaaac6c.png
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给定$n$，答案就是$1$楼的第$1$个数列，该数列的递推式是$2$楼或$17$楼。
而我需要的是通项公式。
17# KeyTo9_Fans
刚才试了下把方程组做z变换，发现可以得到两组解，解的结构比我想象的要简洁。
可惜做逆z变换的时候，Mathematica罢工了。
b=b=b=b=c=c=c=c=1;
b=c=2;
d=d=d=d=0;
d=1;
ZTransform[{
a==Sum(c-c),{i,0,n-1}],
b==1+Sumd,{i,0,n-1}],
c==c+Sumc,{i,0,n}],
d==d+c
},n,z]
附件是 a的其中一个解，
如果能得到该式子的逆z变换的表达，此题就可以破了：
\frac{z^6}{2 (z-2)}-\frac{3 z^5}{z-2}+\frac{6 z^4}{z-2}-\frac{9 z^3}{2 (z-2)}-z^3+\frac{z^2}{z-2}+z^2-\frac{\sqrt{z^4-6 z^3+9 z^2-4} z^4}{2 (z-2)}+\frac{3 \sqrt{z^4-6 z^3+9 z^2-4} z^3}{2 (z-2)}-\frac{\sqrt{z^4-6 z^3+9 z^2-4} z^2}{2 (z-2)}
用Mathematica自带的DifferenceRoot函数，可以得到b,c,d的符号表达。
但a不行。b=
UnitStep-2^(n-2) (1-UnitStep) DifferenceRoot+(2-n) y+(-9-9 n) y+(39+21 n) y+(-46-16 n) y+(16+4 n) y==0,y[-3]==0,y[-2]==0,y[-1]==0,y==0,y==1}]]=====================c=
UnitStep-1/2 (1-UnitStep) DifferenceRoot+(8-4 n) y+(-9-9 n) y+(24+15 n) y+(-19-7 n) y+(4+n) y==0,y==0,y==1,y==-2,y==-2,y==-2}]]=====================d=
1/2 UnitStep (1+DifferenceRoot-4 (-2+n) (2+n) (3+n) y-9 n (1+n) (3+n) y+3 n (19+21 n+5 n^2) y-n (2+n) (18+7 n) y+n (2+n) (3+n) y==0,y==0,y==-3,y==-3,y==-3,y==-11}]]-DifferenceRoot+(8-4 n) y+(-9-9 n) y+(24+15 n) y+(-19-7 n) y+(4+n) y==0,y==0,y==1,y==-2,y==-2,y==-2}]]-(n-3) DifferenceRoot+(8-4 n) y+(-9-9 n) y+(24+15 n) y+(-19-7 n) y+(4+n) y==0,y==0,y==1,y==-2,y==-2,y==-2}]])
足够细心的话，就可以从DifferenceRoot找到b,c,d各自的递归式子的，:)
囧，好像有问题
是$17#$的递推式有问题，还是Mathematica的结果有问题呢？
我百科了一下Z变换，还是不知道Z变换是什么，怎么用。
wayne大牛如果能说些易懂的话让小弟明白Z变换就再好不过了:loveliness:
#####
运行$16#$的程序，确实输出了$1$楼的第$1$个数列。
$17#$的递推式是照着$16#$的程序写的，只是变量名变了。
#####
解释得不错，是时候学习学习了:loveliness:
25# KeyTo9_Fans
我也不清楚，有时间再排查一下
===============
z变换只是一种积分变换(空间变换)工具，是拉普拉斯变换的一个离散域的版本。
如果一个问题在时域不好解决，我们通过z变换之类的空间变换工具，将它变换到另一个空间(比如这里的频域)，往往会有意想不到的启示(enlightenment，revelation)
===========
举一个例子：
假如wayne某天脑子突然间抽风了，已知地推公式a = 3 a - 2 a ， 却不知道怎么求通项公式a。但他竟然懂得这么做：
1)Z变换，即先将表达式变换到另一个空间里，俗称，将时域变换到频域。
2)逆Z变换，返回到原问题域
3)核查一下结果的正确性
关于Z变换和逆Z变换怎么算出结果的，原则上是根据定义算一下积分，但当我们知道了一些很常见的式子的正反变换，结合Z变换的基本性质，
或者会查Z变换表的话，是可以应对很多一般性的问题的，类似于曾经我们经常翻阅的积分表
==========
好像已经足够了。
so，Fans 就拿本题实战一下吧
25# KeyTo9_Fans
用Z变换的好处就是可以套用卷积定理，(时域中的卷积对应于频域中的乘积)，
能将递推式子里的求和(刚好是卷积的形式) 简化成两个函数的乘积的形式。
17# KeyTo9_Fans
设 P(n)=\sum _{i=0}^n b(i) d (n-i)
则，容易得知：
b(n)=1+P(n)
d(n+3)-2d(n+2)+d(n+1) =P(n+1)-P(n)
26# wayne
请教个问题，已知生成函数GeneratingFunction，如何得到通项公式呢？
我知道Maple里可以用convert(1/(1-2*x),FPS,x);，Mathematica中还不清楚怎么用。
29# chyanog SeriesCoefficient