# 2287. [HZOI 2015] Crazy robot

The question ： Starting from the origin , go n Time , Every time up and down, left and right , Only in the first quadrant , Finally, back to the origin, the number of solutions

That's not to mention , The combination number is written to find convolution NTT, And then I didn't think about the first quadrant gg

In fact, that is Carter LAN number

It's just here \(C(i)\) It's No \(\frac{i}{2}\) term , The odd number is 0

Make \(f[n]\) To go n The number of times to go back to the origin ,$$

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

Pay attention to factorial and factorial inverse, don't multiply wrong , Don't lose anything ！！！

```cpp

#include <iostream>

#include <cstdio>

#include <cstring>

#include <algorithm>

#include <cmath>

using namespace std;

typedef long long ll;

const int N=(1<<18)+5, INF=1e9;

const ll P=998244353;

inline int read(){

char c=getchar();int x=0,f=1;

while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}

while(c>='0'&&c<='9'){x=x*10+c-'0';c=getchar();}

return x*f;

}

ll Pow(ll a, ll b, ll P) {

ll ans=1;

for(; b; b>>=1, a=a*a%P)

if(b&1) ans=ans*a%P;

return ans;

}

namespace NTT{

int n, rev[N], g;

void ini(int lim) {

g=3;

n=1; int k=0;

while(n<lim) n<<=1, k++;

for(int i=0; i<n; i++) rev[i] = (rev[i>>1]>>1) | ((i&1)<<(k-1));

}

void dft(ll *a, int flag) {

for(int i=0; i<n; i++) if(i<rev[i]) swap(a[i], a[rev[i]]);

for(int l=2; l<=n; l<<=1) {

int m=l>>1;

ll wn=Pow(g, flag==1 ? (P-1)/l : P-1-(P-1)/l, P);

for(ll *p=a; p!=a+n; p+=l) {

ll w=1;

for(int k=0; k<m; k++) {

ll t = w*p[k+m]%P;

p[k+m] = (p[k] - t + P)%P;

p[k] = (p[k] + t)%P;

w = w*wn%P;

}

}

}

if(flag==-1) {

ll inv=Pow(n, P-2, P);

for(int i=0; i<n; i++) a[i] = a[i]*inv%P;

}

}

void mul(ll *a, ll *b) {

dft(a, 1);

for(int i=0; i<n; i++) a[i]=a[i]*a[i]%P;

dft(a, -1);

}

}using NTT::dft; using NTT::ini; using NTT::mul;

int n;

ll inv[N], fac[N], facInv[N];

ll a[N], b[N];

ll C(int n, int m) {return fac[n]*facInv[m]%P*facInv[n-m]%P;}

int main() {

//freopen("in","r",stdin);

freopen("crazy_robot.in","r",stdin);

freopen("crazy_robot.out","w",stdout);

n=read(); ini(n+n+1);

inv[1]=1; fac[0]=facInv[0]=1;

for(int i=1; i<=n; i++) {

if(i!=1) inv[i] = (P-P/i)*inv[P%i]%P;

fac[i] = fac[i-1]*i%P;

facInv[i] = facInv[i-1]*inv[i]%P;

}

a[0]=b[0]=1;

for(int i=2; i<=n; i+=2) a[i]=b[i]= C(i, i>>1) * inv[(i>>1)+1] %P * facInv[i] %P;

mul(a, b);

for(int i=0; i<=n; i++) a[i]=a[i]*fac[i]%P;

ll ans=0;

for(int m=0; m<=n; m+=2) (ans += C(n, m) * a[m]%P) %=P;

printf("%lld\n", ans);

}

```\]

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