LOJ6703题解

为什么能把微分算子丢到多项式里面去算啊

考虑一个十分简单的 DP：

\[dp[i][j]=dp[i-1][j]+(a_i+j)\times dp[i-1][j-1] \]

考虑把这玩意儿写成 GF，但是会发现求导套了一层 \(x\) 不是很好推。
于是考虑令 \(f[i][j]=dp[i][i-j]\)：

\[f[i][j]=f[i-1][j-1]+(a_i+i-j)\times dp[i-1][j] \]

令 \(F_i(x)=\sum f[i][j]x^j\)，有：

\[F_n(x)=(a_n+n+x)\times F_{n-1}(x)-x\frac{{\rm d}}{{\rm d}x}F_{n-1}(x) \]

令 \(b_i=a_i+i\)，有：

\[F_n(x)=(b_n+x-x\frac{{\rm d}}{{\rm d}x})F_{n-1}(x) \]

那么就有

\[F_n(x)=\prod_{i=1}^{n}(b_i+x-x\frac{{\rm d}}{{\rm d}x}) \]

这个 \(\frac{{\rm d}}{{\rm d}x}\) 很麻烦，用多项式复合暂时去掉这玩意儿：

设 \(f(x)=\prod_{i=1}^{n}(b_i+x)\)，答案就是 \(f(x-x\frac{{\rm d}}{{\rm d}x})\) 的所有系数之和减去 \(1\)。（有一个空子序列）

求 \(f\) 很容易，考虑怎么求 \((x-x\frac{{\rm d}}{{\rm d}x})^k\)。

设 \(G_k(x)=(x-x\frac{{\rm d}}{{\rm d}x})^k\)，容易得到：

\[G_k(x)=(x-x\frac{{\rm d}}{{\rm d}x})G_{k-1}(x) \]

\[G_k[n]=G_{k-1}[n-1]-n\times G_{k-1}[n] \]

如果将这个 \(-1\) 看做一个权值，不难得到 \(G_k[n]=\begin{Bmatrix}k\\n\end{Bmatrix}(-1)^{k-n}\)。

考虑怎么求 \(B[n]=\sum_{i=0}^{n}\begin{Bmatrix}n\\i\end{Bmatrix}(-1)^{n-i}\)：

\[B[n]=\sum_{i=0}^{n}(-1)^{n-i}\frac{1}{i!}\sum_{j=0}^{i}\binom{j}{i}(-1)^{i-j}j^n \]

\[B[n]=(-1)^{n}\sum_{i=0}^{n}\frac{1}{i!}\sum_{j=0}^{i}\binom{j}{i}(-1)^{j}[x^n]e^{jx} \]

\[B[n]=[x^n](-1)^{n}\sum_{i=0}^{n}\frac{1}{i!}(1-e^x)^i \]

\[B[n]=(-1)^n[x^n]\exp(1-e^x) \]

\[B[n]=[x^n]\exp(1-e^{-x}) \]

跑一个 \(\exp\) 和分治乘就好了。复杂度 \(O(n\log^2n)\)。

#include<algorithm>
#include<cstdio>
#include<cctype>
#include<vector>
#define IMP(lim,act) for(int qwq=(lim),i=0;i^qwq;++i)act
const int M=3e5+5,mod=998244353;
int Inv[M],buf[M<<2];int*now=buf,*w[23];
inline void swap(int&a,int&b){
	int c=a;a=b;b=c;
}
inline int Add(const int&a,const int&b){
	return a+b>=mod?a+b-mod:a+b;
}
inline int Del(const int&a,const int&b){
	return b>a?a-b+mod:a-b;
}
inline int max(const int&a,const int&b){
	return a>b?a:b;
}
inline void write(int n){
	static char s[10];int top(0);while(s[++top]=n%10^48,n/=10);while(putchar(s[top--]),top);
}
inline int read(){
	int n(0);char s;while(!isdigit(s=getchar()));while(n=n*10+(s&15),isdigit(s=getchar()));return n;
}
inline int pow(int a,int b=mod-2){
	int ans(1);for(;b;b>>=1,a=1ull*a*a%mod)if(b&1)ans=1ull*ans*a%mod;return ans;
}
inline int Getlen(const int&n){
	int len(0);while((1<<len)<n)++len;return len;
}
inline void init(const int&n){
	const int&m=Getlen(n);w[m]=now;now+=1<<m;Inv[1]=1;
	for(int i=2;i<n;++i)Inv[i]=1ull*(mod-mod/i)*Inv[mod%i]%mod;
	w[m][0]=1;w[m][1]=pow(3,mod-1>>m+1);for(int i=2;i<(1<<m);++i)w[m][i]=1ull*w[m][i-1]*w[m][1]%mod;
	for(int k=m-1;k>=0&&(w[k]=now,now+=1<<k);--k)IMP(1<<k,w[k][i]=w[k+1][i<<1]);
}
struct Poly{
	std::vector<int>F;
	Poly(const Poly&G){F=G.F;}
	Poly(const std::vector<int>G){F=G;}
	Poly(const int&x=0){if(x)F=std::vector<int>(x);}
	inline Poly&resize(const int&len){
		F.resize(len);return*this;
	}
	inline int size()const{
		return F.size();
	}
	inline int&operator[](const int&id){
		return F[id];
	}
	inline void push_back(const int&x){
		F.push_back(x);
	}
	inline Poly&reverse(){
		std::reverse(F.begin(),F.end());return*this;
	}
	inline Poly operator>>(const int&x){
		int i;Poly G;IMP(F.size()-x,G.push_back(F[i+x]));return G;
	}
	inline Poly operator<<(const int&x){
		int i;Poly G;G.resize(x);IMP(F.size(),G.push_back(F[i]));return G;
	}
	inline int operator()(const int&x){
		int i,y(1),ans(0);IMP(F.size(),ans=(ans+1ull*F[i]*y)%mod),y=1ull*y*x%mod;return ans;
	}
	inline void px(Poly G){
		F.resize(max(F.size(),G.size()));G.resize(F.size());
		for(int i(0);i^F.size();++i)F[i]=1ull*F[i]*G[i]%mod;
	}
	inline Poly&Der(){
		for(int i(1);i^F.size();++i)F[i-1]=1ull*F[i]*i%mod;F.pop_back();return*this;
	}
	inline Poly&Int(){
		F.push_back(0);
		for(int i(F.size()-1);i;--i)F[i]=1ull*F[i-1]*::Inv[i]%mod;F[0]=0;return*this;
	}
	inline void DFT(const int&M){
		int i,k,d,x,y,len,*W,*L,*R;F.resize(1<<M);
		for(len=F.size()>>1,d=M-1;len;--d,len>>=1)for(k=0;k^F.size();k+=len<<1){
			W=w[d];L=&F[k];R=&F[k|len];IMP(len,(x=*L,y=*R)),*L++=Add(x,y),*R++=1ull**W++*Del(x,y)%mod;
		}
	}
	inline void IDFT(const int&M){
		int i,k,d,x,y,len,*W,*L,*R;F.resize(1<<M);
		for(len=1,d=0;len^F.size();len<<=1,++d)for(k=0;k^F.size();k+=len<<1){
			W=w[d];L=&F[k];R=&F[k|len];IMP(len,(x=*L,y=1ull**W++**R%mod)),*L++=Add(x,y),*R++=Del(x,y);
		}
		k=::pow(F.size());IMP(F.size(),F[i]=1ull*F[i]*k%mod);for(i=1;(i<<1)<F.size();++i)swap(F[i],F[F.size()-i]);
	}
	inline Poly operator+(Poly G)const{
		Poly F=this->F;int i;F.resize(max(F.size(),G.size()));G.resize(F.size());
		IMP(F.size(),F[i]=Add(F[i],G[i]));return F;
	}
	inline Poly operator-(Poly G)const{
		Poly F=this->F;int i;F.resize(max(F.size(),G.size()));G.resize(F.size());
		IMP(F.size(),F[i]=Del(F[i],G[i]));return F;
	}
	inline Poly operator*(const int&x)const{
		Poly F=this->F;int i;
		IMP(F.size(),F[i]=1ull*F[i]*x%mod);return F;
	}
	inline Poly operator*(Poly G)const{
		Poly F=*this;const int&m=F.size()+G.size()-1,&len=Getlen(m);
		F.DFT(len);G.DFT(len);F.px(G);F.IDFT(len);return F.resize(m);
	}
	inline Poly operator/(Poly G){
		Poly F=*this,sav;const int&m=F.size()-G.size()+1;
		sav.resize(m);IMP(m,sav[i]=G.size()+i<m?0:G[G.size()-m+i]);
		sav.reverse().inv();sav*=F.reverse();
		return sav.resize(m).reverse();
	}
	inline Poly operator%(Poly G){
		return(*this-*this/G*G).resize(G.size()-1);
	}
	inline Poly&inv(){
		Poly b1,b2,b3;const int&m=Getlen(F.size());if(!F.empty())b1.push_back(::pow(F[0]));
		for(int len=1;len<=m;++len){
			b3=b1*2;(b2=F).resize(1<<len);
			b1.DFT(len+1);b1.px(b1);b2.DFT(len+1);b1.px(b2);b1.IDFT(len+1);
			b1=b3-b1.resize(1<<len);
		}
		return*this=b1.resize(F.size());
	}
	inline Poly&ln(){
		const int&m=F.size()-1;Poly G=*this;return(this->Der()*=G.inv()).resize(m).Int();
	}
	inline Poly&exp(){
		Poly b1,b2,b3;const int&m=Getlen(F.size());b1.push_back(1);
		for(int len=1;len<=m;++len){
			b3=b2=b1;b2.resize(1<<len).ln();b2=(*this-b2).resize(1<<len);++b2[0];
			b2.DFT(len);b3.DFT(len);b2.px(b3);b2.IDFT(len);b1.resize(1<<len);
			IMP(1<<len-1,b1[1<<len-1|i]=b2[1<<len-1|i]);
		}
		return*this=b1.resize(F.size());
	}
	inline Poly&sqrt(){
		Poly b1,b2;const int&m=Getlen(F.size());b1.push_back(1);
		for(int len=1;len<=m;++len){
			b2=(b1*2).resize(1<<len).inv();
			b1.DFT(len);b1.px(b1);b1.IDFT(len);
			b1=((*this+b1).resize(1<<len)*b2).resize(1<<len);
		}
		return*this=b1.resize(F.size());
	}
	inline Poly&pow(const int&k){
		int i;ln();IMP(F.size(),F[i]=1ull*F[i]*k%mod);return exp();
	}
	inline Poly&operator>>=(const int&x){
		return*this=operator>>(x);
	}
	inline Poly&operator<<=(const int&x){
		return*this=operator<<(x);
	}
	inline Poly&operator+=(const Poly&G){
		return*this=*this+G;
	}
	inline Poly&operator-=(const Poly&G){
		return*this=*this-G;
	}
	inline Poly&operator*=(const Poly&G){
		return*this=*this*G;
	}
	inline Poly&operator/=(const Poly&G){
		return*this=*this/G;
	}
	inline Poly&operator%=(const Poly&G){
		return*this=*this%G;
	}
};
inline Poly resize(Poly F,const int&n){
	return F.resize(n);
}
inline Poly reverse(Poly F){
	return F.reverse();
}
inline Poly Int(Poly F){
	return F.Int();
}
inline Poly Der(Poly F){
	return F.Der();
}
inline Poly px(Poly F,Poly G){
	return F.px(G),F;
}
inline Poly inv(Poly F){
	return F.inv();
}
inline Poly ln(Poly F){
	return F.ln();
}
inline Poly exp(Poly F){
	return F.exp();
}
inline Poly sqrt(Poly F){
	return F.sqrt();
}
inline Poly pow(Poly F,const int&k){
	return F.pow(k);
}
int n,a[M],fac[M],ifac[M];Poly F,G;
inline Poly DFS(const int&L,const int&R){
	static Poly F;if(L==R)return std::vector<int>().swap(F.F),F.push_back(a[L]),F.push_back(1),F;
	const int&mid=L+R>>1;return DFS(L,mid)*DFS(mid+1,R);
}
signed main(){
	int sum(0);n=read();init(n+1<<1);for(int i=1;i<=n;++i)a[i]=(read()+i)%mod;F=DFS(1,n);
	fac[0]=ifac[0]=1;for(int i=1;i<=n;++i)fac[i]=1ll*fac[i-1]*i%mod,ifac[i]=1ll*ifac[i-1]*Inv[i]%mod;
	G.resize(n+1);for(int i=1;i<=n;++i)G[i]=i&1?ifac[i]:mod-ifac[i];G.exp();
	for(int i=0;i<=n;++i)sum=(sum+1ll*F[i]*G[i]%mod*fac[i])%mod;write((sum-1)%mod);
}