CF322F

CF322F
拉格朗日插值

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
#define M 1100001
#define LL long long 
#define P ((LL)1e9+7)
using namespace std;
 
int n,m,k;
LL f[M],inv[M],Inp[M],Inl[M],A=1,res;
 
LL qick(LL x,LL y)
{
	LL z=1;
	for(y;y;y>>=1, x=x*x%P) if(y&1) z=z*x%P;
	return z;
}
 
int main()
{
	scanf("%d%d",&n,&m);
	k=m+2;
 
	for(int i=1;i<=k;i++) f[i]=(f[i-1]+qick(i,m))%P;
	inv[0]=Inp[0]=inv[1]=Inp[1]=1; 
	for(int i=2;i<=k;i++) inv[i]=(P-P/i)*inv[P%i]%P;
	for(int i=1;i<=k;i++) Inp[i]=Inp[i-1]*inv[i]%P;
	for(int i=n;i>=n-k;i--) A=A*i%P;
	if(k>=n) {printf("%I64d",f[n]); return 0;}
	for(int i=1;i<=k;i++)
	{
		int w=qick(n-i,P-2);
		LL Inv=(Inp[i]*Inp[k-i])%P; 
		if((k-i)&1) Inv*=-1;
		res=(res+Inv*A%P*w%P*f[i])%P;
	}	
	printf("%I64d",(res+P)%P);
}