G47 斯特林反演

视频链接：https://www.bilibili.com/video/BV1ZM411476C/

Luogu P6620 [省选联考 2020 A 卷] 组合数问题

#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;
#define N 1006
int n, x, P, m;
int a[N],S[N][N],J[N],pw[N],pw1[N];

int qpow(int a, int b){
  int res=1;
  while(b){
      if(b&1)res=1ll*res*a%P;
      a=1ll*a*a%P; b>>=1;
  }
  return res;
}
int main(){
  cin >> n >> x >> P >> m;
  for(int i=0; i<=m; ++i)scanf("%d",a+i);
  
  S[0][0]=1;
  for(int i=1; i<=m; ++i)
      for(int j=1; j<=i; ++j)
        S[i][j]=(S[i-1][j-1]+1ll*j*S[i-1][j]%P)%P;
  J[0]=1;
  for(int i=1; i<=m; ++i)
      J[i]=1ll*J[i-1]*(n-i+1)%P;
  for(int i=0; i<=m; ++i)
      pw[i]=qpow(x, i), pw1[i]=qpow(x+1, n-i);
      
  int ans=0;
  for(int i=0; i<=m; ++i){
      int sum=0;
      for(int j=0; j<=i; ++j)
        sum=(sum+1ll*S[i][j]*J[j]%P*pw[j]%P*pw1[j])%P;
      ans=(ans+1ll*a[i]*sum)%P;
  }
  cout << ans << endl;
}