【校內模擬】親（二項式展開）（多項式快速冪）（快速插值）（MTT）

簡要題意：

你有一個數，初始爲$0$，有 $n$ 個機會，每個機會有 $Q/(1+Q)$ 的概率使你的數 +1，請計算所有小於等於你的數的自然數的 $k$ 次冪之和的期望。

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題解：

設 $f_{n,i}$ 表示用了前 $n$ 個機會，你的數的 $i$ 次冪的期望。

轉移考慮二項式展開，乘的東西不變，直接多項式快速冪。

根據期望的線性性，由於 $k$ 次冪之和是個 $k+1$ 次多項式，插值求出它的係數表達然後線性相加即可。

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代碼：

#include<bits/stdc++.h>
#define ll long long
#define re register
#define cs const

using std::cerr;
using std::cout;
#define fi first
#define se second

cs int mod=1e9+7;
inline int add(int a,int b){return a+b>=mod?a+b-mod:a+b;}
inline int dec(int a,int b){return a-b<0?a-b+mod:a-b;}
inline int mul(int a,int b){ll r=(ll)a*b;return r>=mod?r%mod:r;}
inline void Inc(int &a,int b){a+=b-mod;a+=a>>31&mod;}
inline void Dec(int &a,int b){a-=b;a+=a>>31&mod;}
inline void Mul(int &a,int b){a=mul(a,b);}
inline int po(int a,int b){int r=1;for(;b;b>>=1,Mul(a,a))if(b&1)Mul(r,a);return r;}
inline void ex_gcd(int a,int b,int&x,int &y){
	if(!b){x=1,y=0;return;}ex_gcd(b,a%b,y,x);y-=a/b*x;
}inline int Inv(int a){int x,y;ex_gcd(mod,a,y,x);return x+(x>>31&mod);}

cs int bit=19,SIZE=1<<bit|7;

int fac[SIZE],ifc[SIZE],inv[SIZE];

void init_fac(){
	fac[0]=fac[1]=1;
	ifc[0]=ifc[1]=1;
	inv[0]=inv[1]=1;
	for(int re i=2;i<SIZE;++i){
		fac[i]=mul(fac[i-1],i);
		inv[i]=mul(inv[mod%i],mod-mod/i);
		ifc[i]=mul(ifc[i-1],inv[i]);
	}
}

namespace MTT{

cs double PI=acos(-1),PI2=PI+PI;
struct cp{
	double x,y;cp(){}
	cp(double _x,double _y):x(_x),y(_y){}
	friend cp operator+(cs cp &a,cs cp &b){return cp(a.x+b.x,a.y+b.y);}
	friend cp operator-(cs cp &a,cs cp &b){return cp(a.x-b.x,a.y-b.y);}
	friend cp operator*(cs cp &a,cs cp &b){return cp(a.x*b.x-a.y*b.y,a.x*b.y+a.y*b.x);}
	cp& operator+=(cs cp &b){x+=b.x,y+=b.y;return *this;}
	cp& operator-=(cs cp &b){x-=b.x,y-=b.y;return *this;}
	cp& operator*=(cs cp &b){return *this=*this*b;}
	cp conj()cs{return cp(x,-y);}
};
inline cp omega(int i,int k){
	return cp(cos(PI2*i/k),sin(PI2*i/k));
}

int r[SIZE];cp *w[bit+1];

void init_omega(){
	for(int re i=1;i<=bit;++i)
		w[i]=new cp[1<<(i-1)];
	for(int re d=1;d<=bit;++d){
		cp wn=omega(1,1<<d);
		for(int re i=0;i<(1<<(d-1));++i)
			w[d][i]=(i&31)?w[d][i-1]*wn:omega(i,1<<d);
	}
}

void DFT(cp *A,int l){
	for(int re i=0;i<l;++i)
	if(i<r[i])std::swap(A[i],A[r[i]]);
	for(int re i=1,d=1;i<l;i<<=1,++d)
	for(int re j=0;j<l;j+=i<<1)
	for(int re k=0;k<i;++k){
		cp &t1=A[j+k],&t2=A[i+j+k];
		cp t=t2*w[d][k];t2=t1-t,t1+=t;
	}
}
void init_len(int l){
	for(int re i=1;i<l;++i)
		r[i]=r[i>>1]>>1|((i&1)?l>>1:0);
}
void mul(int *a,int *b,int l,int *c){
	static cp A[SIZE],B[SIZE],C[SIZE],D[SIZE];
	for(int re i=0;i<l;++i){
		A[i]=cp(a[i]&0x7fff,a[i]>>15);
		B[i]=cp(b[i]&0x7fff,b[i]>>15);
	}init_len(l),DFT(A,l),DFT(B,l);
	for(int re i=0;i<l;++i){
		int u=(l-i)&(l-1);
		C[i]=(A[i].conj()+A[u])*cp(.5,0)*B[u];
		D[i]=(A[i].conj()-A[u])*cp(0,.5)*B[u];
	}DFT(C,l);DFT(D,l);
	for(int re i=0;i<l;++i){
		ll x=C[i].x/l+.5,y=(C[i].y+D[i].x)/l+.5,z=D[i].y/l+.5;
		x%=mod,y%=mod,z%=mod;c[i]=(x+(y<<15)%mod+(z<<30)%mod)%mod;
	}
}

}

typedef std::vector<int> Poly;
Poly operator*(Poly a,int b){
	for(size_t re i=0;i<a.size();++i)
		Mul(a[i],b);return a;
}Poly operator+(cs Poly &a,cs Poly &b){
	Poly c=a;if(c.size()<b.size())c.resize(b.size());
	for(size_t re i=0;i<b.size();++i)Inc(c[i],b[i]);
	return c;
}Poly operator-(cs Poly &a,cs Poly &b){
	Poly c=a;if(c.size()<b.size())c.resize(b.size());
	for(size_t re i=0;i<b.size();++i)Dec(c[i],b[i]);
	return c;
}Poly operator*(Poly a,Poly b){
	if(!a.size()||!b.size())
		return Poly(0);
	int deg=a.size()+b.size()-1;
	int l=1;while(l<deg)l<<=1;
	a.resize(l),b.resize(l);
	MTT::mul(&a[0],&b[0],l,&a[0]);
	return Poly(a.begin(),a.begin()+deg);
}Poly Inv(cs Poly &a,int lim){
	int n=a.size();Poly c,b(1,Inv(a[0]));
	for(int re l=4;(l>>2)<lim;l<<=1){
		c.resize(l>>1);
		for(int re i=0;i<(l>>1);++i)
			c[i]=i<n?a[i]:0;
		b=b*2-c*b*b;b.resize(l>>1);
	}return Poly(b.begin(),b.begin()+lim);
}Poly Integ(Poly a){
	if(!a.size())return Poly(0);
	a.push_back(0);
	for(size_t re i=a.size()-1;i;--i)
		a[i]=mul(a[i-1],inv[i]);
	a[0]=0;return a;
}Poly Deriv(Poly a){
	if(!a.size())return Poly(0);
	for(size_t re i=1;i<a.size();++i)
		a[i-1]=mul(a[i],i);
	a.pop_back();return a;
}Poly Ln(Poly a,int lim){
	a=Deriv(a)*Inv(a,lim);
	a.resize(lim+1);return Integ(a);
}Poly Exp(cs Poly &a,int lim){
	int n=a.size();Poly c,b(1,1);
	for(int re i=2;(i>>1)<lim;i<<=1){
		c=Ln(b,i);Dec(c[0],1);
		for(int re j=0;j<i;++j)
			c[j]=dec(j<n?a[j]:0,c[j]);
		b=b*c;b.resize(i);
	}return Poly(b.begin(),b.begin()+lim);
}

std::pair<Poly,Poly> interpolation(int l,int r,cs Poly &A,cs Poly &coef){
	if(l==r){
		Poly h(2,0);h[0]=dec(0,l),h[1]=1;
		Poly g(1,mul(A[l],coef[l]));
		return std::pair<Poly,Poly>(h,g);
	}int mid=(l+r)>>1;
	std::pair<Poly,Poly> a=interpolation(l,mid,A,coef);
	std::pair<Poly,Poly> b=interpolation(mid+1,r,A,coef);
	return std::pair<Poly,Poly>(a.fi*b.fi,a.fi*b.se+b.fi*a.se);
}Poly interpolation(cs Poly &G){
	int n=G.size();Poly coef(n);
	for(int re i=0;i<n;++i)
		coef[i]=mul(ifc[i],((n-i)&1)?ifc[n-i-1]:mod-ifc[n-i-1]);
	return interpolation(0,n-1,G,coef).se;
}

Poly calc(int k){//f(x)=\sum_{i=0}^x i^k
	Poly f(k+2);
	for(int i=0,sm=0;i<=k+1;++i)
		f[i]=sm,Inc(sm,po(i+1,k));
	return interpolation(f);
}

int n,k,Q;
void Main(){
	MTT::init_omega();init_fac();
	scanf("%d%d%d",&n,&k,&Q);Poly f(k+2);int iv=Inv(Q+1);
	for(int re i=0;i<=k+1;++i)f[i]=mul(mul(Q,iv),ifc[i]);
	Inc(f[0],iv);f=Ln(f,k+2);int vl=po(Q+1,n);
	for(int re i=0;i<=k+1;++i)Mul(f[i],n);
	f=Exp(f,k+2);Poly g=calc(k);int ans=0;
	for(int re i=0;i<=k+1;++i)
		Inc(ans,mul(mul(mul(f[i],vl),fac[i]),g[i]));
	cout<<ans<<"\n";
}

inline void file(){
#ifdef zxyoi
	freopen("qin.in","r",stdin);
#else
	freopen("qin.in","r",stdin);
	freopen("qin.out","w",stdout);
#endif
}signed main(){file();Main();return 0;}