codeforces 622f

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第一道拉格朗日插值

#include<cstdio>#include<cstdlib>#include<cstring>#include<iostream>#include<algorithm>using namespace std;typedef long long LL;const int maxn = 1000006;const LL mod = 1e9 + 7;int n, k;LL po[maxn];//prod_i(1~maxn)LL ne[maxn];//prod_i (-1~-maxn)LL c[maxn];//系数 po[i-1]*ne[i-k-2]LL S, Y[maxn];LL x, y;LL ex_gcd(LL num1, LL num2){if (!num2){x = 1;y = 0;return num1;}LL ans = ex_gcd(num2, num1%num2);LL tmp = x;x = y;y = tmp - (num1 / num2)*x;return ans;}LL inv(LL xx){ex_gcd(xx, mod);return (x % mod + mod) % mod;}void pre(){po[0] = ne[0] = 1;for (int i = 1; i <= k + 2; i++){int x = mod - i;po[i] = (po[i - 1] * i) % mod;ne[i] = (ne[i - 1] * x) % mod;}for (int i = 1; i <= k + 2; i++){c[i] = (inv(po[i - 1]) * inv(ne[k + 2 - i])) % mod;}S = 1;for (int i = n - k - 2; i <= n - 1; i++){S = (S*i) % mod;}}LL pow(LL a, int k){LL ret = 1;while (k){if (k & 1) ret = (ret * a) % mod;a = (a * a) % mod;k >>= 1;}return ret;}int main(){cin >> n >> k;if (!k){cout << n << endl;return 0;}for (int i = 1; i <= min(k + 2, n); i++){Y[i] = (Y[i - 1] + pow(i*1LL, k)) % mod;}if (n <= k + 2){printf("%d\n", Y[n]);return 0;}pre();LL ret = 0;for (int i = 1; i <= k + 2; i++){LL mid = (((c[i] * S) % mod) * Y[i])%mod;ret = (ret + mid * inv(n - i)) % mod;}cout << ret << endl;return 0;}

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