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

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

   
   
   
   

    
    
    
    1
2
3
10
   
   
   
   

 

Sample Output

   
   
   
   

    
    
    
    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