[组合数学]Catalan数

http://www.51nod.com/Challenge/Problem.html#!#problemId=1120
写到这题的时候突然恍然大悟Catalan数到底有啥用了。就像出栈序列，要是有元素想出栈，那栈内必须要有元素。再比如，机器人走方格不能走到对角线以下，那就要求往右走的要恒大于等于往下走的。 就像出栈序列，入栈的总个数一定要恒大于等于出栈的总个数。
OEIS: Catalan numbers: C(n) = binomial(2n, n) / (n + 1) = (2n)! / (n! (n + 1)!)
Catalan数递推式: C(n) = C(n-1) * (4*n – 2) / (n + 1)
Catalan数表：1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304
存个全都写一起的模板吧我太懒了

ll fac[maxn] = {1};
ll C(ll n, ll m){return n < m ? 0 : (fac[n] * quickpow((fac[m] * fac[n - m]) % mod, mod - 2)) % mod;}
ll lucas(ll n, ll m){return m == 0 ? 1LL : C(n % mod, m % mod) * lucas(n / mod, m / mod) % mod;}
ll rev(ll z, ll m){return z * quickpow(m, mod - 2) % mod;}

顺便来个求Catalan数的Java大数模板

import java.math.BigInteger;
import java.util.Scanner;

public class Main {
    public static void main(String args[]) {
        Scanner inScanner = new Scanner(System.in);
        BigInteger array[] = new BigInteger[105];
        array[1] = BigInteger.valueOf(1);
        for (int i = 2; i <= 100; ++i) {
            array[i] = (array[i - 1].multiply(BigInteger.valueOf(4 * i - 2))).divide(BigInteger.valueOf(i + 1));
        }
        while (inScanner.hasNext()) {
            int n = inScanner.nextInt();
            System.out.println(array[n]);
        }
        inScanner.close();
    }
}