HDU 1023 Train Problem II Catalan数(卡特兰数)
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HDU 1023 Train Problem II
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
Problem Description
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.
Input
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.
Output
For each test case, you should output how many ways that all the trains can get out of the railway.
Sample Input
12310
Sample Output
12516796HintThe 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
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