Codeforces 932E Team Work

题目大意

链接：CF932E

求\(\sum\limits_{i=0}^n\binom nii^k\)。

题目分析

\[\begin{split} \sum_{i=0}^n\binom nii^k&=\sum_{i=0}^n\binom ni\sum_{j=0}^k\binom ij\begin{Bmatrix}k\\j\end{Bmatrix}j!\\ &=\sum_{j=0}^k\begin{Bmatrix}k\\j\end{Bmatrix}j!\sum_{i=0}^n\binom ni\binom ij\\ \end{split} \]

根据常识，有

\[\binom ni\binom ij=\binom nj\binom {n-j}{n-i} \]

可得

\[\begin{split} \sum_{j=0}^k\begin{Bmatrix}k\\j\end{Bmatrix}j!\sum_{i=0}^n\binom ni\binom ij&=\sum_{j=0}^k\begin{Bmatrix}k\\j\end{Bmatrix}j!\sum_{i=0}^n\binom nj\binom {n-j}{n-i}\\ &=\sum_{j=0}^k\begin{Bmatrix}k\\j\end{Bmatrix}j!\binom nj\sum_{i=0}^n\binom {n-j}{n-i}\\ &=\sum_{j=0}^k\begin{Bmatrix}k\\j\end{Bmatrix}j!\binom nj\sum_{i=0}^{n-j}\binom {n-j}i\\ &=\sum_{j=0}^k\begin{Bmatrix}k\\j\end{Bmatrix}j!\binom nj\cdot 2^{n-j} \end{split} \]

代码实现

#include<iostream>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<cstdio>
#include<iomanip>
#include<cstdlib>
#define MAXN 0x7fffffff
typedef long long LL;
const int N=5005,mod=1e9+7;
using namespace std;
inline int Getint(){register int x=0,f=1;register char ch=getchar();while(!isdigit(ch)){if(ch=='-')f=-1;ch=getchar();}while(isdigit(ch)){x=x*10+ch-'0';ch=getchar();}return x*f;}
int s[N][N];
int ksm(int x,int k){
	int ret=1;
	while(k){
		if(k&1)ret=(LL)ret*x%mod;
		x=(LL)x*x%mod,k>>=1;
	}
	return ret;
}
int fac[N],inv[N];
int C(int n,int m){
	if(n<m)return 0;
	return (LL)fac[m]*inv[m]%mod;
}
int main(){
	int n=Getint(),K=Getint();
	s[0][0]=1;
	for(int i=1;i<N;i++)
		for(int j=1;j<N;j++)
			s[i][j]=((LL)s[i-1][j]*j+s[i-1][j-1])%mod;
	
	fac[0]=1;for(int i=1;i<=K;i++)fac[i]=(LL)fac[i-1]*(n-i+1)%mod;
	inv[0]=1;for(int i=1;i<=K;i++)inv[i]=(LL)inv[i-1]*ksm(i,mod-2)%mod;
	int ans=0;
	for(int i=0,t=1,lim=min(n,K);i<=lim;i++,t=(LL)t*i%mod){
		ans=(ans+(LL)s[K][i]*t%mod*C(n,i)%mod*ksm(2,n-i)%mod)%mod;
	} 
	cout<<(ans+mod)%mod;
	return 0;
}

P.S

升级版：【BZOJ5093】图的价值