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生成函数初练

Luogu P4841

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题意

有标号无向连通图计数

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题解

设无向连通图的指数生成函数为$ F$，无向图的指数生成函数为$ G$

有$ F=Ln(G)$

直接求解就好了

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代码

#include<ctime>
#include<cmath>
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
#include<queue>
#include<vector>
#define rt register int
#define ll long long
using namespace std;
inline ll read(){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 k,m,n,x,y,z,cnt,ans;
namespace poly{#define p 1004535809vector<int>R;vector<int>get(int n){vector<int>ret(n);for(rt i=0;i<n;i++)ret[i]=read();return ret;}void print(const vector<int>A){for(auto i:A)write((i+p)%p),putchar(' ');}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;}void NTT(int n,vector<int>&A,int fla){A.resize(n);for(rt i=0;i<n;i++)if(i>R[i])swap(A[i],A[R[i]]);for(rt i=1;i<n;i<<=1){int w=ksm(3,(p-1)/2/i);for(rt j=0;j<n;j+=i<<1){int K=1;for(rt k=0;k<i;k++,K=1ll*K*w%p){int x=A[j+k],y=1ll*K*A[i+j+k]%p;A[j+k]=(x+y)%p,A[i+j+k]=(x-y)%p;}}}if(fla==-1){reverse(A.begin()+1,A.end());int invn=ksm(n);for(rt i=0;i<n;i++)A[i]=1ll*A[i]*invn%p;}}vector<int>Resize(int n,vector<int>A){A.resize(n);return A;}vector<int>Mul(vector<int>x,vector<int>y){int lim=1,sz=x.size()+y.size()-1;while(lim<=sz)lim<<=1;R.resize(lim);for(rt i=0;i<lim;i++)R[i]=(R[i>>1]>>1)|(i&1)*(lim>>1);NTT(lim,x,1);NTT(lim,y,1);for(rt i=0;i<lim;i++)x[i]=1ll*x[i]*y[i]%p;NTT(lim,x,-1);x.resize(sz);return x;}vector<int>Inv(vector<int>A,int n=-1){if(n==-1)n=A.size();if(n==1)return vector<int>(1,ksm(A[0]));vector<int>b=Inv(A,(n+1)/2);int lim=1;while(lim<=n+n)lim<<=1;R.resize(lim);for(rt i=0;i<lim;i++)R[i]=(R[i>>1]>>1)|(i&1)*(lim>>1);A.resize(n);NTT(lim,A,1);NTT(lim,b,1);for(rt i=0;i<lim;i++)A[i]=1ll*b[i]*(2ll-1ll*A[i]*b[i]%p)%p;NTT(lim,A,-1);A.resize(n);return A;}vector<int>Div(vector<int>A,vector<int>B){int n=A.size(),m=B.size();reverse(A.begin(),A.end());reverse(B.begin(),B.end());A.resize(n-m+1),B.resize(n-m+1);int lim=1;while(lim<=2*(n-m+1))lim<<=1;R.resize(lim);for(rt i=0;i<lim;i++)R[i]=(R[i>>1]>>1)|(i&1)*(lim>>1);vector<int>ans=Resize(n-m+1,Mul(A,Inv(B)));reverse(ans.begin(),ans.end());return ans;}vector<int>Add(vector<int>A,vector<int>B){int len=max(A.size(),B.size());A.resize(len);for(rt i=0;i<len;i++)(A[i]+=B[i])%=p;return A;}vector<int>Sub(vector<int>A,vector<int>B){int len=max(A.size(),B.size());A.resize(len);for(rt i=0;i<len;i++)(A[i]-=B[i])%=p;return A;}vector<int>Mul(int x,vector<int>A){for(rt i=0;i<A.size();i++)A[i]=1ll*A[i]*x%p;return A;}vector<int>deriv(vector<int>A){//求导 for(rt i=1;i<A.size();i++)(A[i-1]=1ll*A[i]*i%p);A.pop_back();return A;}vector<int>integ(vector<int>A){//积分 A.push_back(0);for(rt i=A.size()-2;i>=0;i--)A[i+1]=1ll*A[i]*ksm(i+1)%p;A[0]=0;return A;}vector<int>Ln(const vector<int>A){return integ(Resize(A.size()-1,Mul(deriv(A),Inv(A))));}vector<int>Exp(vector<int>A,int n=-1){if(n==-1)n=A.size();if(n==1)return vector<int>(1,1);vector<int>A0=Resize(n,Exp(A,(n+1)>>1));vector<int>now=Resize(n,Ln(A0));for(rt i=0;i<n;i++)now[i]=(A[i]-now[i])%p;now[0]++;return Resize(n,Mul(A0,now));}struct cp{ll a,b,z;//a+bsqrt(z)cp operator *(const cp s)const{return {(1ll*a*s.a%p+1ll*b*s.b%p*z%p)%p,(1ll*a*s.b%p+1ll*b*s.a)%p,z};}};cp ksm(cp x,int y){cp ans={1,0,x.z};for(rt i=y;i;i>>=1,x=x*x)if(i&1){ans=x*ans;}return ans;}int Sqrt(int n){//求二次剩馀 if(ksm(n,(p-1)/2)!=1)return -1;while(1){x=rand()%p;if(ksm((1ll*x*x%p-n%p+p)%p,(p-1)/2)==1)continue;cp ret=ksm({x,1,(1ll*x*x%p+p-n)%p},(p+1)/2);        return min(ret.a,p-ret.a);}}vector<int>GetSqrt(vector<int>A,int n=-1){if(n==-1)n=A.size();if(n==1)return vector<int>(1,Sqrt(A[0]));vector<int>ans=Resize(n,GetSqrt(A,n+1>>1)),C(A.begin(),A.begin()+n);return Resize(n,Mul(ksm(2),Add(ans,Mul(Inv(ans),C))));}vector<int>Pow(vector<int>A,int k){A[0]=1;return Exp(Mul(k,Ln(A)));}//#undef p
}; 
using namespace poly;
vector<int>G,F;
int jc[130010],njc[130010],inv[130010];
int main(){n=read();G.resize(n+1);for(rt i=0;i<=1;i++)jc[i]=njc[i]=inv[i]=1;for(rt i=2;i<=n;i++){jc[i]=1ll*jc[i-1]*i%p;inv[i]=1ll*inv[p%i]*(p-p/i)%p;njc[i]=1ll*njc[i-1]*inv[i]%p;}for(rt i=0;i<=n;i++)G[i]=1ll*ksm(2,1ll*i*(i-1)/2%(p-1))*njc[i]%p;F=Ln(G);cout<<(1ll*F[n]*jc[n]%p+p)%p;return 0;
}