# BZOJ 5093[Lydsy1711月赛]图的价值 线性做法

### 幂与第二类斯特林数

$$x^k=\sum_{j=0}^kj!\binom{x}{j}\begin{Bmatrix}k\\j\end{Bmatrix}$$

$$\begin{Bmatrix}k\\j\end{Bmatrix}=\frac{1}{j!}\sum_{i=0}^ji^k\binom{j}{i}(-1)^{j-i}$$

### 推式子

\begin{aligned} f(n,k) & =\sum_{i=0}^n\binom{n}{i}i^k=\sum_{j=0}\begin{Bmatrix}k\\j\end{Bmatrix}\frac{n!}{(n-j)!}2^{n-j}\\ & =\sum_{j=0}^k\frac{n!}{(n-j)!}2^{n-j}\frac{1}{j!}\sum_{i=0}^j(-1)^{j-i}\binom{j}{i}i^k\\ &=\sum_{j=0}^k\binom{n}{j}2^{n-j}\sum_{i=0}^j(-1)^{j-i}\binom{j}{i}i^k\\ &=\sum_{i=0}^k\binom{n}{i}i^k\sum_{j=i}^k2^{n-j}(-1)^{j-i}\binom{n-i}{j-i}\\ &=\sum_{i=0}^k\binom{n}{i}i^k2^{n-i}\sum_{j=0}^{k-i}\binom{n-i}{j}(-\frac{1}{2})^j \end{aligned}

\begin{aligned} \sum_{j=0}^{k-i}\binom{n-i}{j}(-\frac{1}{2})^j&=\sum_{j=0}^{k-i}\left(\binom{n-i-1}{j}+\binom{n-i-1}{j-1}\right)(-\frac{1}{2})^j\\ &=\sum_{j=1}^{k-i}(-\frac{1}{2})^j\binom{n-i-1}{j-1}+\sum_{j=0}^{k-i}(-\frac{1}{2})^j\binom{n-i-1}{j}\\ &=-\frac{1}{2}\sum_{j=0}^{k-i-1}(-\frac{1}{2})^j\binom{n-i-1}{j}+\sum_{j=0}^{k-i}(-\frac{1}{2})^j\binom{n-i-1}{j}\\ &=\frac{1}{2}\sum_{j=0}^{k-i-1}(-\frac{1}{2})^j\binom{n-i-1}{j}+(-\frac{1}{2})^{k-i}\binom{n-i-1}{k-i} \end{aligned}

### 大常数代码

#include<ctime>
#include<cmath>
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#include<queue>
#define p 998244353
#define inv2 499122177
#define rt register int
#define ll long long
using namespace std;
ll x = 0; char zf = 1; char ch = getchar();
while (ch != '-' && !isdigit(ch)) ch = getchar();
if (ch == '-') zf = -1, ch = getchar();
while (isdigit(ch)) x = x * 10 + ch - '0', ch = getchar(); return x * zf;
}
void write(ll y){if(y<0)putchar('-'),y=-y;if(y>9)write(y/10);putchar(y%10+48);}
void writeln(const ll y){write(y);putchar('\n');}
int i,j,k,m,n,x,y,z,cnt;
int ksm(int x,int y=p-2){
int ans=1;
for(rt i=y;i;i>>=1,x=1ll*x*x%p)if(i&1)ans=1ll*ans*x%p;
return ans;
}
int inv[500010],A[500010];
int v[500010],ss[500010];bool b[500010];
int main(){
inv[0]=inv[1]=1;
v[1]=1;v[0]=(k==0);
for(rt i=2;i<=k;i++){
if(!b[i])ss[++cnt]=i,v[i]=ksm(i,k);
for(rt j=1;i*ss[j]<=k&&j<=cnt;j++){
b[i*ss[j]]=1;v[i*ss[j]]=1ll*v[i]*v[ss[j]]%p;
if(i%ss[j]==0)break;
}
}
for(rt i=2;i<=k;i++)inv[i]=1ll*inv[p%i]*(p-p/i)%p;
int ans=0;
if(n<=k){
for(rt i=0,c=1;i<=n;c=1ll*(n-i)*inv[i+1]%p,i++)
ans+=1ll*c*v[i]%p;
cout<<(1ll*ans*(n+1)%p*ksm(2,(ll)n*(n-1)/2%(p-1))%p+p)%p;
return 0;
}

A[k]=1;
for(rt i=k-1,y=-inv2,c=n-i-1;i>=0;i--,y=1ll*y*-inv2%p){
A[i]=(1ll*A[i+1]*inv2%p+1ll*c*y%p)%p;
c=1ll*c*(n-i)%p*inv[k-i+1]%p;
}

for(rt i=0,d=ksm(2,n),c=1;i<=k&&i<=n;c=1ll*c*(n-i)%p*inv[i+1]%p,i++,d=1ll*d*inv2%p)
(ans+=1ll*c*v[i]%p*d%p*A[i]%p)%=p;
cout<<(1ll*ans*(n+1)%p*ksm(2,(ll)n*(n-1)/2%(p-1))%p+p)%p;
return 0;
}

posted @ 2019-03-05 20:29  Kananix  阅读(348)  评论(6编辑  收藏