G75 拉插 CF622F The Sum of the k-th Powers

视频链接：G75 拉插 CF622F The Sum of the k-th Powers_哔哩哔哩_bilibili

 

 

 

 

The Sum of the k-th Powers - 洛谷 | 计算机科学教育新生态

// 拉插 O(klogM)
#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;

#define LL long long
const int N=1e6+10,M=1e9+7;
int n,k;
LL pre[N],suf[N],fac[N],y,a,b,ans;

LL ksm(LL a,LL b){
  LL s=1;
  while(b){
    if(b&1)s=s*a%M;
    a=a*a%M;
    b>>=1;
  }
  return s;
}
int main(){
  scanf("%d%d",&n,&k);
  pre[0]=suf[k+3]=fac[0]=1;
  for(int i=1;i<=k+2;i++)
    pre[i]=pre[i-1]*(n-i)%M;//前缀积
  for(int i=k+2;i>=1;i--)
    suf[i]=suf[i+1]*(n-i)%M;//后缀积
  for(int i=1;i<=k+2;i++)
    fac[i]=fac[i-1]*i%M;    //阶乘
  for(int i=1;i<=k+2;i++){
    y=(y+ksm(i,k))%M;
    a=pre[i-1]*suf[i+1]%M;
    b=fac[i-1]*fac[k+2-i]*((k+2-i)&1?-1:1)%M;
    ans=(ans+y*a%M*ksm(b,M-2)%M)%M;
  }
  printf("%d\n",(ans+M)%M);
}

 

// 拉插 O(klogk)
#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;

#define LL long long
const int N=1e6+10,M=1e9+7;
int n,k;
LL pre[N],suf[N],fac[N],inv[N],y,a,b,ans;

LL ksm(LL a,LL b){
  LL s=1;
  while(b){
    if(b&1)s=s*a%M;
    a=a*a%M;
    b>>=1;
  }
  return s;
}
int main(){
  scanf("%d%d",&n,&k);
  pre[0]=suf[k+3]=fac[0]=inv[0]=1;
  for(int i=1;i<=k+2;i++)
    pre[i]=pre[i-1]*(n-i)%M;//前缀积
  for(int i=k+2;i>=1;i--)
    suf[i]=suf[i+1]*(n-i)%M;//后缀积
  for(int i=1;i<=k+2;i++)
    fac[i]=fac[i-1]*i%M;    //阶乘
  inv[k+2]=ksm(fac[k+2],M-2);
  for(int i=k+1;i>=1;i--)
    inv[i]=inv[i+1]*(i+1)%M; //逆元
  
  for(int i=1;i<=k+2;i++){
    y=(y+ksm(i,k))%M;
    a=pre[i-1]*suf[i+1]%M;
    b=inv[i-1]*inv[k+2-i]*((k+2-i)&1?-1:1)%M;
    ans=(ans+y*a%M*b%M)%M;
  }
  printf("%d\n",(ans+M)%M);
}