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\(Solution\)
Sakika 简式Ko:
\ [F (n-) = \ sum_ {I} = 0 n-^ \ sum_ {J} = 0 ^ I \ bmatrix the begin {I} \\ J \ bmatrix End {} * 2 ^ J * J! \]
\ [ f (n) = \ sum_ {
j = 0} ^ n2 ^ j * j! \ sum_ {i = 0} ^ n \ begin {bmatrix} i \\ j \ end {bmatrix} \] The second type of Manchester formulas of the forest:
\ [F (n-) = \ sum_ {J = 0} ^ N2 ^ J * J \ sum_ {I = 0} ^ n-\ sum_ {K = 0} ^ J \ FRAC {(-. 1! ) ^ K} {K!} * \ FRAC {(JK) ^ I} {(JK)!} \]
\ [F (n-) = \ sum_ {J = 0} ^ N2 ^ J * J! \ sum_ { k = 0} ^ j \ frac
! {! (- 1) ^ k} {k} * \ frac {\ sum_ {i = 0} ^ n (jk) ^ i} {(jk)} \] we let \ (F (i) = \ frac
{(- 1) ^ k} {! i}, G (i) = \ frac {\ sum_ {j = 0} ^ ni ^ j} {! i} \) The geometric series summation formula can be obtained:
\ (\ sum_ {J} = 0 ^ J ^ Ni = \ {I ^ {n-FRAC. 1} + {-1}}. 1-I \)
so \ (G (i) = \ {I ^ {n-FRAC. 1} + {-1} (. 1-I) * I!} \)
\ (G (0) =. 1, G (. 1). 1 = n-+ \)
then
\ [f (n) = \ sum_ {j = 0}
^ n2 ^ j * j! \ sum_ {k = 0} ^ jF (k) * G (jk) \] because when \ (j> k \) when \ (G (jk) = 0 \)
so that the original formula is equivalent to:
\ [f (n) = \
sum_ {j = 0} ^ n2 ^ j * j! \ sum_ {k = 0} ^ nF (k) * G (jk) \] This thing directly \ (NTT \) engage in a do just fine
\(Code\)
#include<bits/stdc++.h>
#define int long long
#define rg register
#define file(x) freopen(x".in","r",stdin);freopen(x".out","w",stdout);
using namespace std;
const int mod=998244353;
const int N=500010;
int read(){
int x=0,f=1;
char c=getchar();
while(c<'0'||c>'9') f=(c=='-')?-1:1,c=getchar();
while(c>='0'&&c<='9') x=x*10+c-48,c=getchar();
return f*x;
}
int f[N],g[N],r[N],limit=1,w[N],inv[N],a[N],b[N],jc[N];
int ksm(int a,int b){
int ans=1;
while(b){
if(b&1) ans=ans*a%mod;
a=a*a%mod,b>>=1;
}
return ans;
}
void ntt(int *a,int opt){
for(int i=0;i<=limit;i++)
if(i<r[i])
swap(a[i],a[r[i]]);
for(int i=1;i<limit;i<<=1){
int w=ksm(3,(mod-1)/(i*2));
if(opt==-1) w=ksm(w,mod-2);
for(int j=0;j<limit;j+=i<<1){
int l=1;
for(int k=j;k<j+i;k++){
int p=l*a[k+i]%mod;
a[k+i]=(a[k]-p+mod)%mod;
a[k]=(a[k]+p)%mod;
l=l*w%mod;
}
}
}
}
main(){
int n=read(),l=0,ans=0;
while(limit<=(n<<1))
limit<<=1,l++;
jc[0]=1;
for(int i=1;i<=limit;i++)
jc[i]=jc[i-1]*i%mod;
inv[limit]=ksm(jc[limit],mod-2);
for(int i=limit-1;i>=0;i--)
inv[i]=inv[i+1]*(i+1)%mod;
g[0]=1,g[1]=n+1,f[1]=mod-1,f[0]=1;
for(int i=2;i<=n;i++)
g[i]=(ksm(i,n+1)-1)%mod*inv[i]%mod*ksm(i-1,mod-2)%mod,f[i]=i&1?mod-inv[i]:inv[i];
for(int i=0;i<limit;i++)
r[i]=(r[i>>1]>>1)|((i&1)<<(l-1));
ntt(f,1),ntt(g,1);
for(int i=0;i<limit;i++)
f[i]=f[i]*g[i]%mod;
ntt(f,-1);
int inv=ksm(limit,mod-2);
for(int i=0;i<=n;i++)
ans=(ans+ksm(2,i)*jc[i]%mod*f[i]%mod*inv%mod)%mod;
printf("%lld",ans);
}