HDU 1023 Train Problem II (卡特兰数)未解决未解决未解决未解决

Train Problem II

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)

Total Submission(s): 7876 Accepted Submission(s): 4223



[align=left]Problem Description[/align]
As we all know the Train Problem I, the boss of the Ignatius Train Station want to know if all the trains come in strict-increasing order, how many orders that all the trains can get out of the railway.

[align=left]Input[/align]
The input contains several test cases. Each test cases consists of a number N(1<=N<=100). The input is terminated by the end of file.

[align=left]Output[/align]
For each test case, you should output how many ways that all the trains can get out of the railway.

[align=left]Sample Input[/align]

1
2
3
10

[align=left]Sample Output[/align]

1
2
5
16796

Hint
The result will be very large, so you may not process it by 32-bit integers.

卡特兰数，又名卡塔兰数，是组合数学里面经常用的一个计数数列。

令h(0)=1,h(1)=1，catalan数满足递推式

h(n)= h(0)*h(n-1)+h(1)*h(n-2) + ... + h(n-1)h(0) (n>=2)
例如：h(2)=h(0)*h(1)+h(1)*h(0)=1*1+1*1=2
h(3)=h(0)*h(2)+h(1)*h(1)+h(2)*h(0)=1*2+1*1+2*1=5
另类递推式

h(n)=h(n-1)*(4*n-2)/(n+1);
递推关系的解为：
h(n)=C(2n,n)/(n+1) (n=0,1,2,...)
递推关系的另类解为：
h(n)=c(2n,n)-c(2n,n-1)(n=0,1,2,...)
（from baike）

等我