拉格朗日插值

O（n）求：

大佬的題解：TP

#include<bits/stdc++.h>
#define R register
#define fp(i,a,b) for(R int i=a,I=b+1;i<I;++i)
#define fd(i,a,b) for(R int i=a,I=b-1;i>I;--i)
#define go(u) for(int i=head[u],v=e[i].v;i;i=e[i].nx,v=e[i].v)
using namespace std;
const int N=1e6+5,P=1e9+7;
inline int add(R int x,R int y){return x+y>=P?x+y-P:x+y;}
inline int dec(R int x,R int y){return x-y<0?x-y+P:x-y;}
inline int mul(R int x,R int y){return 1ll*x*y-1ll*x*y/P*P;}

int ksm(R int x,R int y){
    R int res=1;
    for(;y;y>>=1,x=mul(x,x))if(y&1)res=mul(res,x);
    return res;
}

int f[N],inv[N];
int n,k;
inline int Inv(R int x){return x<=k?inv[x]:ksm(x,P-2);}
int Large(int k,int n){
    if(k<=n)return f[k];
    int ty=(n&1)?P-1:1,tmp=1,res=0;
    fp(i,1,n)tmp=1ll*tmp*(k-i)%P*Inv(i)%P;
    fp(i,0,n){
        res=add(res,1ll*f[i]*tmp%P*ty%P);
        tmp=1ll*tmp*(k-i)%P*Inv(k-i-1)%P*(n-i)%P*Inv(i+1)%P;
        ty=P-ty;
    }
    return res;
}
int main(){
//  freopen("testdata.in","r",stdin);
    scanf("%d%d",&n,&k);
    inv[0]=inv[1]=1;fp(i,2,k)inv[i]=1ll*inv[P%i]*(P-P/i)%P;
    fp(i,1,k+1)f[i]=add(f[i-1],ksm(i,k));
    printf("%d\n",Large(n,k+1));
    return 0;
}