結果

問題 No.1080 Strange Squared Score Sum
コンテスト
ユーザー lam6er
提出日時 2025-04-09 20:56:20
言語 PyPy3
(7.3.15)
結果
TLE  
実行時間 -
コード長 2,084 bytes
コンパイル時間 169 ms
コンパイル使用メモリ 82,912 KB
実行使用メモリ 82,924 KB
最終ジャッジ日時 2025-04-09 20:58:18
合計ジャッジ時間 7,308 ms
ジャッジサーバーID
（参考情報） 		judge2 / judge1
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ファイルパターン 結果
sample AC * 2
other TLE * 1 -- * 19
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ソースコード

diff # 

import sys
MOD = 10**9 + 9

def main():
    N = int(sys.stdin.readline().strip())
    if N == 0:
        return
    
    # Precompute factorial and inverse factorial modulo MOD
    fact = [1] * (N + 1)
    for i in range(1, N + 1):
        fact[i] = fact[i-1] * i % MOD
    
    # Compute g(x) = sum_{a=1 to N} (a+1)^2 x^a
    g = [0] * (N+1)
    for a in range(1, N+1):
        if a > N:
            continue
        g[a] = (a + 1) ** 2 % MOD
    
    # Now compute cos(g) + sin(g) - 1 mod x^(N+1)
    # Initialize C (cos) and S (sin)
    C = [0] * (N+1)
    S = [0] * (N+1)
    C[0] = 1  # cos(0) = 1
    # S[0] = 0  # sin(0) = 0
    
    # Precompute the derivative g'(x)
    gp = [0] * (N)
    for a in range(0, N):
        gp[a] = ( (a+1) * ( (a+2) ** 2 ) ) % MOD
    
    # Compute C and S using dynamic programming based on their differential equations
    for k in range(1, N+1):
        # Compute C[k] using C' = -g' * S
        # The coefficient of x^{k-1} in C' is given by the convolution of gp and S up to x^{k-1}
        # C' = -gp * S
        conv = 0
        for i in range(max(0, k-1 - (N-1)), k):
            if i >= len(gp):
                continue
            j = (k-1) - i
            if j < 0 or j > N:
                continue
            conv += gp[i] * S[j]
            conv %= MOD
        Ck = (-conv) % MOD
        Ck = Ck * pow(k, MOD-2, MOD) % MOD
        C[k] = Ck
        
        # Compute S[k] using S' = gp * C
        conv = 0
        for i in range(max(0, k-1 - (N-1)), k):
            if i >= len(gp):
                continue
            j = (k-1) - i
            if j < 0 or j > N:
                continue
            conv += gp[i] * C[j]
            conv %= MOD
        Sk = conv * pow(k, MOD-2, MOD) % MOD
        S[k] = Sk
    
    # Now compute S_total = C + S - 1
    S_total = [ (C[i] + S[i]) % MOD for i in range(N+1) ]
    S_total[0] = (S_total[0] - 1) % MOD
    if N >= 0:
        S_total[0] %= MOD
    
    for k in range(1, N+1):
        res = S_total[k] * fact[N] % MOD
        print(res)
    
if __name__ == "__main__":
    main()
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