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传送门：【51nod】算法马拉松4 F 移数字
涉及知识点：多项式求逆，多项式除法，多点插值，阶乘取模。
对于N!%P，复杂度为 O(N−−√log2N−−√) 。
但常数巨大，和暴力算实际复杂度只相差常数= =
这个是可以扩展到组合数取模的~
my code:
#include <stdio.h>
#include <string.h>
#include <map>
#include <math.h>
#include <vector>
#include <algorithm>
using namespace std ;typedef long long LL ;
typedef long long Int ;
#define clr( a , x ) memset ( a , x , sizeof a )
#define ls ( o << 1 )
#define rs ( o << 1 | 1 )
#define lson ls , l , m
#define rson rs , m + 1 , r
#define root 1 , 1 , Sqrt
#define mid ( ( l + r ) >> 1 )const int MAXN = 300005 ;vector < int > M[MAXN << 2] ;
vector < int > F[MAXN << 2] ;
int x1[MAXN] , x2[MAXN] , x3[MAXN] , tmp[MAXN] ;
int A[MAXN] , B[MAXN] , R[MAXN] ;
int a[MAXN] ;
int mod , g ;
int S[MAXN] , top ;
int n ;
int ans ;
int Sqrt ;int exgcd ( int a , int b , int& x , int& y ) {if ( b ) {exgcd ( b , a % b , y , x ) ;y -= a / b * x ;} else {x = 1 ;y = 0 ;}
}int inv ( int a ) {int x , y , b = mod ;exgcd ( a , b , x , y ) ;if ( x < 0 ) x += mod ;return x ;
}int powmod ( int a , int b ) {int res = 1 , tmp = a ;while ( b ) {if ( b & 1 ) res = ( LL ) res * tmp % mod ;tmp = ( LL ) tmp * tmp % mod ;b >>= 1 ;}return res ;
}void DFT ( int y[] , int n , int rev ) {for ( int i = 1 , j , k , t ; i < n ; ++ i ) {for ( j = 0 , k = n >> 1 , t = i ; k ; k >>= 1 , t >>= 1 ) j = j << 1 | t & 1 ;if ( i < j ) swap ( y[i] , y[j] ) ;}for ( int s = 2 , ds = 1 ; s <= n ; ds = s , s <<= 1 ) {int wn = powmod ( g , ( mod - 1 ) / s ) ;if ( rev ) wn = inv ( wn ) ;for ( int k = 0 ; k < n ; k += s ) {LL w = 1 , t ;for ( int i = k ; i < k + ds ; ++ i , w = w * wn % mod ) {y[i + ds] = ( y[i] - ( t = w * y[i + ds] % mod ) + mod ) % mod ;y[i] = ( y[i] + t ) % mod ;}}}
}void INV ( int A[] , int B[] , int n ) {B[0] = inv ( A[0] ) ;int i , n1 , t , vn , s , ds ;for ( s = 2 , ds = 1 ; ds < n ; ds = s , s <<= 1 ) {n1 = ( s << 1 ) , t = min ( s , n ) , vn = inv ( n1 ) ;for ( i = 0 ; i < t ; ++ i ) tmp[i] = A[i] ;for ( i = t ; i < n1 ; ++ i ) tmp[i] = 0 ;DFT ( tmp , n1 , 0 ) ;DFT ( B , n1 , 0 ) ;for ( i = 0 ; i < n1 ; ++ i ) B[i] = B[i] * ( 2 - ( LL ) tmp[i] * B[i] % mod + mod ) % mod ;DFT ( B , n1 , 1 ) ;for ( i = 0 ; i < t ; ++ i ) B[i] = ( LL ) B[i] * vn % mod ;for ( i = t ; i < n1 ; ++ i ) B[i] = 0 ;}
}void DIV ( int A[] , int B[] , int R[] , int n , int m ) {int n1 = 1 , n2 = n - m + 1 , i ;while ( n1 <= n * 2 ) n1 <<= 1 ;for ( i = 0 ; i < n ; ++ i ) x1[i] = A[n - i - 1] ;for ( i = 0 ; i < m ; ++ i ) x2[i] = B[m - i - 1] ;for ( i = m ; i < n2 ; ++ i ) x2[i] = 0 ;for ( i = n2 ; i < n1 ; ++ i ) x1[i] = x2[i] = 0 ;for ( i = 0 ; i < n1 ; ++ i ) x3[i] = 0 ;INV ( x2 , x3 , n2 ) ;DFT ( x1 , n1 , 0 ) ;DFT ( x3 , n1 , 0 ) ;for ( i = 0 ; i < n1 ; ++ i ) x1[i] = ( LL ) x1[i] * x3[i] % mod ;DFT ( x1 , n1 , 1 ) ;int vn = inv ( n1 ) ;for ( i = 0 ; i < n2 ; ++ i ) x2[n2 - i - 1] = ( LL ) x1[i] * vn % mod ;for ( i = n2 ; i < n1 ; ++ i ) x2[i] = 0 ;for ( i = m ; i < n1 ; ++ i ) B[i] = 0 ;DFT ( x2 , n1 , 0 ) ;DFT ( B , n1 , 0 ) ;for ( i = 0 ; i < n1 ; ++ i ) x2[i] = ( LL ) x2[i] * B[i] % mod ;DFT ( x2 , n1 , 1 ) ;for ( i = 0 ; i < m - 1 ; ++ i ) {R[i] = A[i] - ( LL ) x2[i] * vn % mod ;if ( R[i] < 0 ) R[i] += mod ;}
}void preprocess ( int n ) {top = 0 ;int i , flag ;for ( i = 2 ; i * i <= n ; ++ i ) {if ( n % i == 0 ) {S[top ++] = i ;while ( n % i == 0 ) n /= i ;}}if ( n > 1 ) S[top ++] = n ;for ( g = 1 ; ; ++ g ) {flag = 1 ;for ( i = 0 ; i < top ; ++ i ) {if ( powmod ( g , ( mod - 1 ) / S[i] ) == 1 ) {flag = 0 ;break ;}}if ( flag ) return ;}
}void deal ( vector < int > & F , vector < int > & F1 , vector < int > & F2 , int sz ) {int n = F1.size () , m = F2.size () , n1 = 1 , i ;while ( n1 < n + m ) n1 <<= 1 ;for ( i = 0 ; i < n ; ++ i ) x1[i] = F1[i] ;for ( i = 0 ; i < m ; ++ i ) x2[i] = F2[i] ;for ( i = n ; i < n1 ; ++ i ) x1[i] = 0 ;for ( i = m ; i < n1 ; ++ i ) x2[i] = 0 ;DFT ( x1 , n1 , 0 ) ;DFT ( x2 , n1 , 0 ) ;for ( i = 0 ; i < n1 ; ++ i ) x1[i] = ( LL ) x1[i] * x2[i] % mod ;DFT ( x1 , n1 , 1 ) ;LL vn = inv ( n1 ) ;for ( i = 0 ; i < sz ; ++ i ) F.push_back ( x1[i] * vn % mod ) ;
}void brute_deal ( vector < int > & F , vector < int > & F1 , vector < int > & F2 , int sz ) {int n = F1.size () , m = F2.size () , i , j ;for ( i = 0 ; i < sz ; ++ i ) x1[i] = 0 ;for ( i = 0 ; i < n ; ++ i ) {for ( j = 0 ; j < m ; ++ j ) {x1[i + j] = ( x1[i + j] + ( LL ) F1[i] * F2[j] ) % mod ;}}for ( i = 0 ; i < sz ; ++ i ) F.push_back ( x1[i] ) ;
}void build ( int o , int l , int r ) {if ( l == r ) {M[o].push_back ( ( mod - a[l] ) % mod ) ;M[o].push_back ( 1 ) ;F[o].push_back ( l ) ;F[o].push_back ( 1 ) ;return ;}int m = mid , n = r - l + 2 ;build ( lson ) ;build ( rson ) ;if ( n <= 1400 ) {brute_deal ( M[o] , M[ls] , M[rs] , n ) ;brute_deal ( F[o] , F[ls] , F[rs] , n ) ;return ;}deal ( M[o] , M[ls] , M[rs] , n ) ;deal ( F[o] , F[ls] , F[rs] , n ) ;
}void get ( int A[] , vector < int > & F , int n ) {for ( int i = 0 ; i < n ; ++ i ) A[i] = F[i] ;
}void go ( int o , int l , int r ) {int m = mid , i , j ;int n = r - l + 2 , nL = F[ls].size () , nR = F[rs].size () ;get ( A , F[o] , n ) ;if ( n <= 500 ) {for ( i = l ; i <= r ; ++ i ) {LL x = 0 , y = 1 ;for ( j = 0 ; j < n ; ++ j ) {x = ( x + A[j] * y ) % mod ;y = y * a[i] % mod ;}ans = ans * x % mod ;}return ;}get ( B , M[ls] , nL ) ;DIV ( A , B , R , n , nL ) ;for ( i = 0 ; i < nL ; ++ i ) F[ls][i] = R[i] ;F[ls][nL - 1] = 0 ;get ( B , M[rs] , nR ) ;DIV ( A , B , R , n , nR ) ;for ( i = 0 ; i < nR ; ++ i ) F[rs][i] = R[i] ;F[rs][nR - 1] = 0 ;go ( lson ) ;go ( rson ) ;
}void calc ( int n ) {Sqrt = sqrt ( 1.0 * n ) ;for ( int i = 0 ; i < Sqrt ; ++ i ) a[i + 1] = Sqrt * i % mod ;build ( root ) ;
// printf ( "ok\n" ) ;go ( root ) ;for ( int i = Sqrt * Sqrt + 1 ; i <= n ; ++ i ) ans = ( LL ) ans * i % mod ;
}void solve () {scanf ( "%d%d" , &n , &mod ) ;if ( n >= mod ) {printf ( "0\n" ) ;return ;}ans = 1 ;preprocess ( mod - 1 ) ;calc ( n ) ;if ( n & 1 ) ans = ( LL ) ans * inv ( 2 ) % mod ;printf ( "%d\n" , ans ) ;
}int main () {solve () ;return 0 ;
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