Codeforces 1549E / 1548C

原题链接:https://codeforces.com/contest/1549/problem/E

题目大意

给定 $n$,$q$ 次询问。

每次询问给定一个 $x$,求 $\sum_{i=0}^n \binom{3i}{x}$

思路

考虑定义 $f_{i,j}$ 表示 $\sum_{k=0}^{n-1} \binom{3k + j}{i}$,令 $0 \leq j < m$

注意到

  1. $\sum_{j=0}^{m-1} f_{i,j} = \sum_{j=0}^{3n – 1} \binom{j}{i} = \binom{3n}{i + 1}$
  2. $f_{i,j} = f_{ i – 1, j – 1 } + f_{i, j – 1 }$

发现可以解出 $f_{i,0}$

$O(n)$ 递推即可。

拓展

对于每天有 $m$ 只羊而不是 $3$ 只的情况。

显然可以 $O(nm)$

Code

/*
 * e.cpp
 * Copyright (C) 2021 Woshiluo Luo <[email protected]>
 *
 * 「Two roads diverged in a wood,and I—
 * I took the one less traveled by,
 * And that has made all the difference.」
 *
 * Distributed under terms of the GNU AGPLv3+ license.
 */

#include <cstdio>
#include <cstdlib>
#include <cstring>

#include <algorithm>

typedef long long ll;
typedef unsigned long long ull;

inline bool isdigit( const char cur ) { return cur >= '0' && cur <= '9'; }/*{{{*/
template <class T> 
T Max( T a, T b ) { return a > b? a: b; }
template <class T> 
T Min( T a, T b ) { return a < b? a: b; }
template <class T> 
void chk_Max( T &a, T b ) { if( b > a ) a = b; }
template <class T> 
void chk_Min( T &a, T b ) { if( b < a ) a = b; }
template <typename T>
T read() { 
    T sum = 0, fl = 1; 
    char ch = getchar();
    for (; isdigit(ch) == 0; ch = getchar())
        if (ch == '-') fl = -1;
    for (; isdigit(ch); ch = getchar()) sum = sum * 10 + ch - '0';
    return sum * fl;
}
template <class T> 
T pow( T a, int p ) {
    T res = 1;
    while( p ) {
        if( p & 1 ) 
            res = res * a;
        a = a * a;
        p >>= 1;
    }
    return res;
}/*}}}*/

const int N = 3e6 + 1e5;
const int mod = 1e9 + 7;

struct ModInt {/*{{{*/
    int cur;
    ModInt( int _cur = 0 ) { cur = ( ( ( _cur % mod ) + mod ) % mod ); }

    inline ModInt operator+ ( const ModInt &b ) const { return ( cur + b.cur ) % mod; }
    inline ModInt operator- ( const ModInt &b ) const { return ( ( ( cur - b.cur ) % mod ) + mod ) % mod; }
    inline ModInt operator* ( const ModInt &b ) const { return ( 1LL * cur * b.cur ) % mod; }
    inline ModInt operator/ ( const ModInt &b ) const { return ( 1LL * cur * pow( b, mod - 2 ).cur ) % mod; }

    inline void operator+= ( const ModInt &b ) { (*this) = (*this) + b; }
    inline void operator-= ( const ModInt &b ) { (*this) = (*this) - b; }
    inline void operator*= ( const ModInt &b ) { (*this) = (*this) * b; }
    inline void operator/= ( const ModInt &b ) { (*this) = (*this) / b; }

    inline void output( const char end = '\n' ) { printf( "%d%c", cur, end ); }
};/*}}}*/


ModInt fac[N], rfac[N], dp[N][3];

ModInt choose( int n, int m ) {
    if( n < m ) 
        return 0;
    return fac[n] * rfac[m] * rfac[ n - m ];
}
void init( int n ) {
    fac[0] = 1;
    for( int i = 1; i <= n; i ++ ) 
        fac[i] = fac[ i - 1 ] * i;
    rfac[n] = (ModInt)1 / fac[n];
    for( int i = n - 1; i >= 0; i -- ) 
        rfac[i] = rfac[ i + 1 ] * ( i + 1 );
}

void get_dp( int n ) {
    int n3 = 3 * n;
    dp[0][0] = dp[0][1] = dp[0][2] = n;
    ModInt inv_3 = (ModInt)1 / 3;
    for( int i = 1; i <= n3; i ++ ) {
        ModInt sum = choose( n3, i + 1 );
        dp[i][0] = ( sum - dp[ i - 1 ][0] * (ModInt)2 - dp[ i - 1 ][1] ) * inv_3;
        dp[i][1] = dp[i][0] + dp[ i - 1 ][0];
        dp[i][2] = dp[i][1] + dp[ i - 1 ][1];
    }
}

int main() {
#ifdef woshiluo
    freopen( "e.in", "r", stdin );
    freopen( "e.out", "w", stdout );
#endif
    int n, q;
    n = read<int>(); q = read<int>();
    init( 3 * n ); get_dp(n);
    while( q -- ) {
        int x = read<int>();
        ( dp[x][0] + choose( 3 * n, x ) ).output();
    }
}

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