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写一份\(O(n)\)的代码，作为参考。

这东西坑点比较多。

我的代码当中多开了几个数组，因为~~我太懒了~~便于理解。

Solution

首先，我们找到正解(拉格朗日差值具体就不讲了)。

发现需要维护：

阶乘的连乘的逆元

由于是从\(1\)开始，需要的空间不算大，所以可以\(O(k)\)处理，下一个。

\(\prod\limits_{i=1}^nx-i\)

这个不是逆元，可以求出每一个元素，连乘再求逆元，按套路就可以了。

\(n\leq k+2\)时的答案

为了保证时间复杂度严格为\(O(k)\)，这个应该线性筛。

Code

#include <cstdio>
using namespace std;
const int N = 1000000 + 5, mod = 1000000007;
inline int mul (int a, int b) {return 1LL * a * b % mod;}
inline int add (int a, int b) {return a + b >= mod ? a + b - mod : a + b;}
inline int mns (int a, int b) {return a - b < 0 ? a - b + mod : a - b;}
inline int qpow (int u, int v) {
    int tot = 1, base = u % mod;
    while (v) {
        if (v & 1) tot = mul (tot, base);
        base = mul (base, base);
        v >>= 1;
    }
    return tot;
}
int x, k;
struct DuLiu {
    int pri[N + 5], tot, res[N + 5];
    bool vis[N + 5];
    inline void init () {
        int n = k + 2;
        res[1] = 1;
        for (int i = 2; i <= n; ++i) {
            if (!vis[i]) {
                pri[++tot] = i;
                res[i] = qpow (i, k);
            }
            for (int j = 1; j <= tot; ++j) {
                if (pri[j] * i > n) break;
                vis[i * pri[j]] = 1;
                res[i * pri[j]] = mul (res[i], res[pri[j]]);
                if (i % pri[j] == 0) break;
            }
        }
        for (int i = 2; i <= n; ++i) res[i] = add (res[i - 1], res[i]);
    }
}num;
int fac[N + 5], fac_prod[N + 5], fac_prod_inv[N + 5], extra, mid[N + 5], mid_prod[N + 5], mid_prod_inv[N + 5], mid_inv[N + 5];
inline void init_mid () {
    int n = k + 2;
    for (int i = 1; i <= n; ++i) mid[i] = x - i; mid[0] = x;
    mid_prod[0] = 1; for (int i = 1; i <= n; ++i) mid_prod[i] = mul (mid_prod[i - 1], mid[i]);
    mid_prod_inv[n] = qpow (mid_prod[n], mod - 2);
    for (int i = n - 1; i >= 0; --i) mid_prod_inv[i] = mul (mid_prod_inv[i + 1], mid[i + 1]);
    for (int i = 1; i <= n; ++i) mid_inv[i] = mul (mid_prod_inv[i], mid_prod[i - 1]);
    mid_inv[0] = qpow (x, mod - 2);
}
inline void Lagrange () {
    int ans = 0, n = k + 2;
    for (int i = 1; i <= n; ++i) {
        int tmp = 1, f = 1;
        tmp = mul (tmp, mul (fac_prod_inv[i - 1], i - 2 >= 0 ? fac_prod[i - 1 - 1] : 1));
        tmp = mul (tmp, mul (fac_prod_inv[n - i], n - i - 1 >= 0 ? fac_prod[n - i - 1] : 1));
        if ((n - i) % 2 == 1) f *= -1;
        int liujuakioi = mid_prod[n];
        tmp = mul (tmp, mul (liujuakioi, mid_inv[i]));
        tmp = mul (tmp, num.res[i]);
        if (f > 0) ans = add (ans, tmp);
        else ans = mns (ans, tmp);
    }
    printf ("%d\n", ans);
}
int main () {
    scanf ("%d%d", &x, &k);
    num.init ();
    if (x <= k + 2) {printf ("%d\n", num.res[x]); return 0;}
    fac[0] = 1; for (int i = 1; i <= N; ++i) fac[i] = mul (fac[i - 1], i);
    fac_prod[0] = 1; for (int i = 1; i <= N; ++i) fac_prod[i] = mul (fac_prod[i - 1], fac[i]);
    fac_prod_inv[N] = qpow (fac_prod[N], mod - 2);
    for (int i = N - 1; i >= 0; --i) fac_prod_inv[i] = mul (fac_prod_inv[i + 1], fac[i + 1]);
    init_mid();
    Lagrange ();
    return 0;
}

Conclusion

细节比较多，交了好几次才过。

注意\(Lagrange\)函数里面的变量的使用，易错(实际上是我写得太麻烦，貌似阶乘的单独逆元另外开一个数组更加方便)。

CF622F:The Sum of the k-th Powers

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