[Codeforces 932E]Team Work

Description

题库链接

求

\[\sum_{i=1}^n C(n,i)\times i^k\]

\(1\leq n\leq 10^9, 1\leq k\leq 5000\)

Solution

[BZOJ 5093]图的价值的弱化版。

看公式推导可以戳链接。

Code

#include <bits/stdc++.h>
using namespace std;
const int N = 5000+5, yzh = 1e9+7;

int S[N][N], n, k, inv[N];

int quick_pow(int a, int b) {
    int ans = 1;
    while (b) {
        if (b&1) ans = 1ll*ans*a%yzh;
        b >>= 1, a = 1ll*a*a%yzh;
    }
    return ans;
}
void work() {
    scanf("%d%d", &n, &k);
    inv[0] = inv[1] = S[0][0] = 1;
    for (int i = 2; i <= k; i++) inv[i] = -1ll*yzh/i*inv[yzh%i]%yzh;
    for (int i = 1; i <= k; i++)
        for (int j = 1; j <= i; j++)
            S[i][j] = (1ll*S[i-1][j-1]+1ll*S[i-1][j]*j%yzh)%yzh;
    int ans = 0, C = n, fac = 1;
    for (int i = 1; i <= min(k, n); i++) {
        (ans += 1ll*fac*C%yzh*S[k][i]%yzh*quick_pow(2, n-i)%yzh) %= yzh;
        fac = 1ll*fac*(i+1)%yzh, C = 1ll*C*(n-i)%yzh*inv[i+1]%yzh;
    }
    printf("%d\n", (ans+yzh)%yzh);
}
int main() {work(); return 0; }