bjfu1238 卡特兰数取余

题目就是指定n，求卡特兰数Ca(n)%m。求卡特兰数有递推公式、通项公式和近似公式三种，因为要取余，所以近似公式直接无法使用，递推公式我简单试了一下，TLE。所以只能从通项公式入手。

Ca(n) = (2*n)! / n! / (n+1)!

思想就是把Ca(n)质因数分解，然后用快速幂取余算最后的答案。不过，算n!时如果从1到n依次质因数分解，肯定是要超时的，好在阶乘取余有规律，不断除素因子即可。

最后还是擦边过，可能筛法写得一般吧，也算是题目要求太柯刻。

/*
 * Author    : ben
 */
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <cmath>
#include <ctime>
#include <iostream>
#include <algorithm>
#include <queue>
#include <set>
#include <map>
#include <stack>
#include <string>
#include <vector>
#include <deque>
#include <list>
#include <functional>
#include <numeric>
#include <cctype>
using namespace std;
#ifdef ON_LOCAL_DEBUG
#else
#endif

typedef long long LL;
const int MAXN = 80000;
const int N = 1000001;
bool isPrime[N + 3];//多用两个元素以免判断边界
int pn, pt[MAXN], num[MAXN];

void init_prime_table() {
    memset(isPrime, true, sizeof(isPrime));
    int p = 2, q, del;
    double temp;
    while (p <= N) {
        while (!isPrime[p]) {        p++;        }
        if (p > N) {//已经结束
            break;        }
        temp = (double) p;
        temp *= p;
        if (temp > N)
            break;
        while (temp <= N) {
            del = (int) temp;            isPrime[del] = false;
            temp *= p;        }
        q = p + 1;
        while (q < N) {
            while (!isPrime[q]) {    q++;    }
            if (q >= N) { break;}
            temp = (double) p;
            temp *= q;
            if (temp > N)    break;
            while (temp <= N) {
                del = (int) temp;
                isPrime[del] = false;
                temp *= p;
            }
            q++;
        }
        p++;
    }
    pn = 0;
    for (int i = 2; i <= N; i++) {
        if (isPrime[i]) {
            pt[pn++] = i;
        }
    }
}

int modular_exp(int a, int b, int c) {
    LL res, temp;
    res = 1 % c, temp = a % c;
    while (b) {
        if (b & 1) {
            res = res * temp % c;
        }
        temp = temp * temp % c;
        b >>= 1;
    }
    return (int) res;
}

void getNums(int n) {
    int x = 2 * n;
    int len = pn;
    for (int i = 0; i < len && pt[i] <= x; i++) {
        int r = 0, a = x;
        while (a) {
            r += a / pt[i];
            a /= pt[i];
        }
        num[i] = r;
    }
    x = n;
    for (int i = 0; i < len && pt[i] <= x; i++) {
        int r = 0, a = x;
        while (a) {
            r += a / pt[i];
            a /= pt[i];
        }
        num[i] -= r;
    }
    x = n + 1;
    for (int i = 0; i < len && pt[i] <= x; i++) {
        int r = 0, a = x;
        while (a) {
            r += a / pt[i];
            a /= pt[i];
        }
        num[i] -= r;
    }
}

int main() {
#ifdef ON_LOCAL_DEBUG
    freopen("data.in", "r", stdin);
//    freopen("test.in", "r", stdin);
//    freopen("data.out", "w", stdout);
#endif
    int n, m;
    init_prime_table();
    while (scanf("%d%d", &n, &m) == 2) {
        getNums(n);
        LL ans = 1;
        int n2 = 2 * n;
        for (int i = 0; ans && (i < pn) && pt[i] <= n2; i++) {
            if (num[i] > 0) {
                ans = ans * (LL) modular_exp(pt[i], num[i], m);
                ans %= m;
            }
        }
        printf("%d
", (int)ans);
    }
    return 0;
}