考虑一个排列和那个是一一对应的
的权值是
前
的个数+1
而一个排列
的个数即欧拉数,设为
计算可以利用二项式反演至少
至少钦定组内递增就是斯特林数
整理得
现在只考虑求 ,剩下翻转卷积即可
只用考虑后面怎么求
设
满足
则
然后利用拓展拉格朗日反演得到
求导裂项分母展开
直接快速幂即可
复杂度
注意
等作为分母常数项会为
需要除
求逆后加回去
#include<bits/stdc++.h>
using namespace std;
#define cs const
#define re register
#define pb push_back
#define pii pair<int,int>
#define ll long long
#define y1 shinkle
#define fi first
#define se second
#define bg begin
cs int RLEN=1<<20|1;
inline char gc(){
static char ibuf[RLEN],*ib,*ob;
(ib==ob)&&(ob=(ib=ibuf)+fread(ibuf,1,RLEN,stdin));
return (ib==ob)?EOF:*ib++;
}
inline int read(){
char ch=gc();
int res=0;bool f=1;
while(!isdigit(ch))f^=ch=='-',ch=gc();
while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
return f?res:-res;
}
inline ll readll(){
char ch=gc();
ll res=0;bool f=1;
while(!isdigit(ch))f^=ch=='-',ch=gc();
while(isdigit(ch))res=(res+(res<<2)<<1)+(ch^48),ch=gc();
return f?res:-res;
}
inline char readchar(){
char ch=gc();
while(isspace(ch))ch=gc();
return ch;
}
inline int readstring(char *s){
int top=0;char ch=gc();
while(isspace(ch))ch=gc();
while(!isspace(ch)&&ch!=EOF)s[++top]=ch,ch=gc();
s[top+1]='\0';return top;
}
template<typename tp>inline void chemx(tp &a,tp b){a=max(a,b);}
template<typename tp>inline void chemn(tp &a,tp b){a=min(a,b);}
cs int mod=998244353;
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)?(a-b+mod):(a-b);}
inline int mul(int a,int b){static ll r;r=(ll)a*b;return (r>=mod)?(r%mod):r;}
inline void Add(int &a,int b){a=(a+b)>=mod?(a+b-mod):(a+b);}
inline void Dec(int &a,int b){a=(a<b)?(a-b+mod):(a-b);}
inline void Mul(int &a,int b){static ll r;r=(ll)a*b;a=(r>=mod)?(r%mod):r;}
inline int ksm(int a,int b,int res=1){for(;b;b>>=1,Mul(a,a))(b&1)&&(Mul(res,a),1);return res;}
inline int Inv(int x){return ksm(x,mod-2);}
inline int fix(ll x){x%=mod;return (x<0)?x+mod:x;}
cs int N=200051;
int fac[N],ifac[N];
typedef vector<int> poly;
inline void init_fac(){
fac[0]=ifac[0]=1;
for(int i=1;i<N;i++)fac[i]=mul(fac[i-1],i);
ifac[N-1]=Inv(fac[N-1]);
for(int i=N-2;i;i--)ifac[i]=mul(ifac[i+1],i+1);
}
inline int Cb(int n,int m){return (n<0||m<0||n<0)?0:mul(fac[n],mul(ifac[m],ifac[n-m]));}
namespace Poly{
cs int C=18,M=(1<<C)|5;
int *w[C+1],rev[M],iv[M];
inline void init_rev(int lim){
for(int i=0;i<lim;i++)rev[i]=(rev[i>>1]>>1)|((i&1)*(lim>>1));
}
inline void init_w(){
for(int i=1;i<=C;i++)w[i]=new int[(1<<(i-1))|1];
int wn=ksm(3,(mod-1)/(1<<C));w[C][0]=1;
for(int i=1,l=1<<(C-1);i<l;i++)w[C][i]=mul(w[C][i-1],wn);
for(int i=C-1;i;i--)
for(int j=0,l=1<<(i-1);j<l;j++)w[i][j]=w[i+1][j<<1];
iv[0]=iv[1]=1;
for(int i=2;i<M;i++)iv[i]=mul(mod-mod/i,iv[mod%i]);
}
inline void dft(int *f,int lim){
for(int i=0;i<lim;i++)if(i>rev[i])swap(f[i],f[rev[i]]);
for(int mid=1,l=1,a0,a1;mid<lim;mid<<=1,l++)
for(int i=0;i<lim;i+=mid<<1)
for(int j=0;j<mid;j++)
a0=f[i+j],a1=mul(w[l][j],f[i+j+mid]),f[i+j]=add(a0,a1),f[i+j+mid]=dec(a0,a1);
}
inline void ntt(poly &f,int lim,int kd){
dft(&f[0],lim);
if(kd==-1){
reverse(f.bg()+1,f.bg()+lim);
for(int i=0;i<lim;i++)Mul(f[i],iv[lim]);
}
}
inline poly operator +(poly a,poly b){
if(a.size()<b.size())a.resize(b.size());
for(int i=0;i<b.size();i++)Add(a[i],b[i]);
return a;
}
inline poly operator -(poly a,poly b){
if(a.size()<b.size())a.resize(b.size());
for(int i=0;i<b.size();i++)Dec(a[i],b[i]);
return a;
}
inline poly operator *(poly a,int b){
for(int i=0;i<a.size();i++)Mul(a[i],b);return a;
}
inline poly operator *(poly a,poly b){
if(!a.size()||!b.size())return poly(0);
int deg=a.size()+b.size()-1;
if(a.size()<=32||b.size()<=32){
poly c(deg,0);
for(int i=0;i<a.size();i++)
for(int j=0;j<b.size();j++)
Add(c[i+j],mul(a[i],b[j]));
return c;
}int lim=1;while(lim<deg)lim<<=1;
init_rev(lim);
a.resize(lim),ntt(a,lim,1);
b.resize(lim),ntt(b,lim,1);
for(int i=0;i<lim;i++)Mul(a[i],b[i]);
ntt(a,lim,-1),a.resize(deg);
return a;
}
inline poly Inv(poly a,int deg){
poly b(1,::Inv(a[0])),c;
for(int lim=4;lim<(deg<<2);lim<<=1){
c.resize(lim>>1);init_rev(lim);
for(int i=0;i<(lim>>1);i++)c[i]=(i<a.size()?a[i]:0);
c.resize(lim),ntt(c,lim,1);
b.resize(lim),ntt(b,lim,1);
for(int i=0;i<lim;i++)Mul(b[i],dec(2,mul(b[i],c[i])));
ntt(b,lim,-1),b.resize(lim>>1);
}b.resize(deg);return b;
}
inline poly deriv(poly a){
for(int i=0;i+1<a.size();i++)a[i]=mul(a[i+1],i+1);
a.pop_back();return a;
}
inline poly integ(poly a){
a.pb(0);
for(int i=a.size()-1;i;i--)a[i]=mul(a[i-1],iv[i]);
a[0]=0;return a;
}
inline poly Ln(poly a,int deg){
a=integ(Inv(a,deg)*deriv(a)),a.resize(deg);return a;
}
inline poly Exp(poly a,int deg){
poly b(1,1),c;
for(int lim=2;lim<(deg<<1);lim<<=1){
c=Ln(b,lim);
for(int i=0;i<lim;i++)c[i]=dec(i<a.size()?a[i]:0,c[i]);
Add(c[0],1);
b=b*c,b.resize(lim);
}b.resize(deg);return b;
}
inline poly ksm(poly a,int b,int deg){
return Exp(Ln(a,deg)*b,deg);
}
inline poly Mulx(poly a){
a.pb(0);
for(int i=a.size()-1;i;i--)a[i]=a[i-1];
a[0]=0;return a;
}
inline poly Divx(poly a){
for(int i=0;i+1<a.size();i++)a[i]=a[i+1];
a.pop_back();return a;
}
}
using namespace Poly;
void write(poly a){
for(int i=0;i<a.size();i++)cout<<a[i]<<" ";puts("");
}
inline poly calc_g(int n){
int lm=n+3;
poly f(lm);
for(int i=0;i<f.size();i++)f[i]=ifac[i+1];
poly res1=Divx(Inv(Divx(poly(1,1)-f),lm)*f);
res1.resize(lm);
poly t(2);t[0]=t[1]=1,t=Inv(Divx(Ln(t,lm)),lm);
poly dn=Inv(Divx(poly(1,1)-t),lm),pt=ksm(t,n+1,lm);
poly v1=pt*dn,v2;v1.resize(lm),dn=dn*dn,dn.resize(lm);
pt=pt*deriv(t),pt.resize(lm),v2=pt*dn,v2.resize(lm);
poly res2(n,0);
for(int i=0;i<n;i++)res2[i]=dec(res1[i],mul(iv[n+1],add(mul(n-i+1,v1[i+1]),v2[i+1])));
return res2;
}
int n;
int main(){
#ifdef Stargazer
freopen("lx.in","r",stdin);
#endif
init_w(),init_fac();
n=read();
poly g=calc_g(n);
for(int i=0;i<g.size();i++)Mul(g[i],fac[i]);
g.resize(n);
poly f(n+1);
for(int i=0;i<=n;i++)f[i]=(i&1)?dec(0,ifac[i]):ifac[i];
reverse(f.bg(),f.end());
f=f*g;
for(int i=1;i<=n;i++)cout<<mul(fac[n],mul(ifac[i-1],f[n+i-1]))<<" ";
return 0;
}