题解 | #不可道F题题解（多项式）#
不可道
https://ac.nowcoder.com/acm/contest/49035/F
F题多项式。谢谢出题人，题目出的好，答疑很良心，爱了qwq
#include<iostream>
#include<algorithm>
#include<string.h>
#include<vector>
using namespace std;
// #define debug(x) cout<<"[debug]"#x<<"="<<x<<endl
typedef long long ll;
typedef long double ld;
typedef pair<int,int> pii;
const double eps=1e-8;
const int INF=0x3f3f3f3f;
#ifndef ONLINE_JUDGE
#define debug(...)
#include<debug>
#else
#define debug(...)
#endif
int n,m,k;
namespace NTT
{
typedef vector<ll> Poly;
const int mod=998244353,G=3,Gi=332748118;
const int M=6e5;//TODO
int bit,tot;
int rev[M];
int inv[M];
int init_inv=[](){
inv[0]=inv[1]=1;
for(int i=2;i<M;i++)
inv[i]=1ll*(mod-mod/i)*inv[mod%i]%mod;
return 0;
}();
ll qmi(ll a,ll b,ll p)
{
ll res=1;
while(b)
{
if(b&1)
res=res*a%p;
a=a*a%p;
b>>=1;
}
return res;
}
void NTT(Poly &a,int inv)
{
for(int i=0;i<tot;i++)
if(i<rev[i]) swap(a[i],a[rev[i]]);
for(int mid=1;mid<tot;mid*=2)
{
ll w1=qmi(inv==1?G:Gi,(mod-1)/(2*mid),mod);
for(int i=0;i<tot;i+=mid*2)
{
ll wk=1;
for(int j=0;j<mid;j++,wk=wk*w1%mod)
{
ll x=a[i+j];
ll y=wk*a[i+j+mid]%mod;
a[i+j]=(x+y)%mod;
a[i+j+mid]=(x-y+mod)%mod;
}
}
}
if(inv==-1)//就不用后面除了
{
ll intot=qmi(tot,mod-2,mod);
for(int i=0;i<tot;i++)
{
a[i]=a[i]*intot%mod;
}
}
}
Poly mul(Poly a,Poly b)//deg是系数的数量，所以有0~deg-1次项
{
int deg=(int)a.size()+b.size()-1;
bit=0;
while((1<<bit)<deg) bit++;//至少要系数的数量
tot=1<<bit; //系数项为0~tot-1
for(int i=0;i<tot;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(bit-1));
Poly c(tot);
a.resize(tot),b.resize(tot);
NTT(a,1),NTT(b,1);
for(int i=0;i<tot;i++) c[i]=a[i]*b[i]%mod;
NTT(c,-1);
c.resize(n+k+1);
return c;
}
Poly operator *(Poly a,Poly b)
{
return mul(a,b);
}
Poly operator *(Poly a,int t)
{
Poly res;
for(int i=0;i<a.size();i++) res.push_back(1ll*a[i]*t%mod);
return res;
}
Poly operator *(Poly a,ll t)
{
Poly res;
for(int i=0;i<a.size();i++) res.push_back(a[i]*t%mod);
return res;
}
Poly operator +(Poly a,Poly b)
{
Poly res(a);
res.resize(max(a.size(),b.size()));
for(int i=0;i<b.size();i++) res[i]=(res[i]+b[i])%mod;
return res;
}
Poly operator -(Poly a,Poly b)
{
Poly res(a);
res.resize(max(a.size(),b.size()));
for(int i=0;i<b.size();i++) res[i]=(res[i]-b[i]+mod)%mod;
return res;
}
Poly Inv(Poly &f,int deg)//多项式f对于x^deg的逆元(注意rev[]等数组要开到2*deg的空间级别,f[]要开deg级别)
{
if(deg==1) return Poly(1,qmi(f[0],mod-2,mod));
Poly B=Inv(f,(deg+1)>>1);//上一个逆元
Poly A(f.begin(),f.begin()+deg);
bit=0;
while((1<<bit)<(deg<<1)) bit++;//至少要系数的数量
tot=1<<bit; //系数项为0~tot-1
for(int i=0;i<tot;i++) rev[i]=(rev[i>>1]>>1)|((i&1)<<(bit-1));
A.resize(tot),B.resize(tot);
NTT(A,1),NTT(B,1);
for(int i=0;i<tot;i++)
A[i]=B[i]*(2-A[i]*B[i]%mod+mod)%mod;
NTT(A,-1);
A.resize(deg);
return A;
}
Poly cdq_ntt(int l,int r,vector<Poly> &f)
{
if(l==r) return f[l];
int mid=l+r>>1;
return mul(cdq_ntt(l,mid,f),cdq_ntt(mid+1,r,f));
}
namespace Cipolla
{
int W;
struct cp
{
ll x,y;
cp(ll x=0,ll y=0): x(x),y(y) {}
cp operator+ (const cp _)const{
return {(x+_.x)%mod,(y+_.y)%mod};
}
cp operator* (const cp _)const{
return {(x*_.x%mod+y*_.y%mod*W%mod)%mod,(x*_.y+y*_.x)%mod};
}
};
cp qmi(cp a,int b)
{
cp res=1;
while(b)
{
if(b&1) res=res*a;
a=a*a;
b>>=1;
}
return res;
}
int cipolla(int n)
{
if(qmi(n,(mod-1)>>1).x!=1)
{
return -1;
}
int a=rand()%mod;
while(!a||qmi((1ll*a*a-n+mod)%mod,(mod-1)>>1).x==1)
{
a=rand()%mod;
}
W=(1ll*a*a-n+mod)%mod;
int x=qmi(cp(a,1),(mod+1)>>1).x;
if(x>mod-x) x=mod-x;
return x;
}
}
Poly Sqrt(Poly &A,int deg)
{
if(deg==1)
{
return {Cipolla::cipolla(A[0])};//A[0]=1修改这里即可
}
Poly f0=Sqrt(A,(deg+1)>>1);
f0.resize(deg);
Poly inf0=Inv(f0,deg);
Poly temp(A.begin(),A.begin()+deg);
temp=mul(inf0,temp);
temp.resize(deg);
for(int i=0;i<deg;i++) f0[i]=(f0[i]+temp[i])*inv[2]%mod;
return f0;
}
Poly Deriv(Poly f)//求导
{
for(int i=0;i<f.size();i++) f[i]=f[i+1]*(i+1)%mod;
f.pop_back();
return f;
}
Poly Integ(Poly f)//积分
{
f.push_back(0);
for(int i=f.size()-1;i>=1;i--) f[i]=f[i-1]*inv[i]%mod;
f[0]=0;
return f;
}
Poly Ln(Poly f,int deg)//f[0]=1
{
f=mul(Deriv(f),Inv(f,deg));
f.resize(deg-1);
return Integ(f);
}
Poly Exp(Poly &A,int deg)//A[0]=0，记得至少开deg大小
{
if(deg==1) return {1};
Poly f0=Exp(A,(deg+1)>>1);
Poly temp=f0;
temp.resize(deg);
temp=Ln(temp,deg);
for(int i=0;i<deg;i++) temp[i]=(A[i]-temp[i]+mod)%mod;
temp[0]=(temp[0]+1)%mod;
temp=mul(f0,temp);
temp.resize(deg);
return temp;
}
Poly qmi_ntt(Poly A,int k,int deg=-1)//A(x)^k且A[0]=1
{
if(deg==-1) deg=(A.size()-1)*k+1;
A.resize(deg);
A=Ln(A,deg);
A=A*k;
A=Exp(A,deg);
return A;
}
Poly qmi_ntt(Poly A,string s,int deg=-1)//A(x)^k
{
ll k=0;//mod (p)
ll kk=0;//mod (p-1),即phi(p)
bool ty=false;
for(int i=0;i<s.size();i++)
{
k=(1ll*k*10+(s[i]-'0'));
if(k>=mod) k%=mod,ty=true;
kk=(1ll*kk*10+(s[i]-'0'))%(mod-1);
}
if(deg==-1) deg=(A.size()-1)*k+1;
A.resize(deg);
int n=A.size();
int t=0;
while(t<n&&!A[t]) t++;
if(t==n||ty&&t>=1)
{
return Poly(n,0);
}
ll temp=qmi(A[t],mod-2,mod);
Poly B;
for(int i=t;i<A.size();i++) B.push_back(A[i]*temp%mod);
B=Ln(B,B.size());
B=B*k;
B=Exp(B,B.size());
B.resize(n);
Poly res(n);
temp=qmi(A[t],kk,mod);
for(ll i=1ll*t*k;i<n;i++) res[i]=B[i-1ll*t*k]*temp%mod;
return res;
}
Poly psqmi_ntt(Poly A,int k)
{
Poly res(1,1);
while(k)
{
if(k&1) res=res*A;
A=A*A;
k>>=1;
}
return res;
}
}
using namespace NTT;
const int N=100005;
int fact[N],infact[N];
void init(int n)
{
fact[0]=1;
for(int i=1;i<=n;i++)
{
fact[i]=1ll*fact[i-1]*i%mod;
}
infact[n]=qmi(fact[n],mod-2,mod);
for(int i=n-1;i>=0;i--)
{
infact[i]=1ll*infact[i+1]*(i+1)%mod;
}
}
int main()
{
init(N-1);
scanf("%d%d%d",&n,&m,&k);
Poly A(n+k+1),B(n+k+1),C(n+k+1);
for(int i=0;i<=n;i++)
{
C[i]=infact[i];
if(i<k) A[i]=infact[i];
else B[i-k]=infact[i];
}
Poly Am=qmi_ntt(A,m,n+k+1);
Poly res=qmi_ntt(C,m+1,n+k+1)-Am*C*(m+1)+Am*A*m;
res=res*Inv(B,n+1);
ll ans=res[n+k]*fact[n]%mod*qmi(qmi(m,n,mod),mod-2,mod)%mod;
printf("%lld\n",ans);
}
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