結果
| 問題 | No.665 Bernoulli Bernoulli |
| ユーザー |
 hitonanode
|
| 提出日時 | 2024-12-02 06:13:37 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 709 ms / 2,000 ms |
| コード長 | 8,753 bytes |
| コンパイル時間 | 1,042 ms |
| コンパイル使用メモリ | 98,748 KB |
| 実行使用メモリ | 5,248 KB |
| 最終ジャッジ日時 | 2024-12-02 06:13:50 |
| 合計ジャッジ時間 | 12,923 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge1 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 15 |
ソースコード
#line 1 "utilities/test/kth_power_sum.yuki665.test.cpp"#define PROBLEM "https://yukicoder.me/problems/no/665"#line 2 "utilities/kth_power_sum.hpp"#include <vector>// Computes the sum of the k-th powers// Complexity:// - O(k) per query,// - O(k^2) precomputation (can be reduced to O(k log k) with FFT)template <class MODINT> struct kth_power_sum {// generating function: x / (e^x - 1) + x// NOTE: use B(1) = 1/2 definitionstd::vector<MODINT> bernoulli;kth_power_sum() = default;void expand() {if (bernoulli.empty()) {bernoulli = {1};} else {const int k = bernoulli.size();MODINT x(0);for (int i = 0; i < k; ++i) { x = -x + bernoulli[i] * MODINT::binom(k + 1, i); }bernoulli.push_back(x / (k + 1));}}// Calculate 1^k + 2^k + ... + n^k// Based on Faulhaber's formula:// S_k(n) = 1 / (k + 1) * sum_{j=0}^{k} B_j * C(k + 1, j) * n^(k + 1 - j)template <class T> MODINT prefix_sum(int k, const T &n) {while ((int)bernoulli.size() <= k) expand();MODINT ret(0), np(1);for (int j = k; j >= 0; --j) {np *= n;ret += MODINT::binom(k + 1, j) * bernoulli[j] * np;}return ret / (k + 1);}};#line 2 "modint.hpp"#include <cassert>#include <iostream>#include <set>#line 6 "modint.hpp"template <int md> struct ModInt {using lint = long long;constexpr static int mod() { return md; }static int get_primitive_root() {static int primitive_root = 0;if (!primitive_root) {primitive_root = [&]() {std::set<int> fac;int v = md - 1;for (lint i = 2; i * i <= v; i++)while (v % i == 0) fac.insert(i), v /= i;if (v > 1) fac.insert(v);for (int g = 1; g < md; g++) {bool ok = true;for (auto i : fac)if (ModInt(g).pow((md - 1) / i) == 1) {ok = false;break;}if (ok) return g;}return -1;}();}return primitive_root;}int val_;int val() const noexcept { return val_; }constexpr ModInt() : val_(0) {}constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }constexpr ModInt(lint v) { _setval(v % md + md); }constexpr explicit operator bool() const { return val_ != 0; }constexpr ModInt operator+(const ModInt &x) const {return ModInt()._setval((lint)val_ + x.val_);}constexpr ModInt operator-(const ModInt &x) const {return ModInt()._setval((lint)val_ - x.val_ + md);}constexpr ModInt operator*(const ModInt &x) const {return ModInt()._setval((lint)val_ * x.val_ % md);}constexpr ModInt operator/(const ModInt &x) const {return ModInt()._setval((lint)val_ * x.inv().val() % md);}constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }constexpr bool operator<(const ModInt &x) const {return val_ < x.val_;} // To use std::map<ModInt, T>friend std::istream &operator>>(std::istream &is, ModInt &x) {lint t;return is >> t, x = ModInt(t), is;}constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {return os << x.val_;}constexpr ModInt pow(lint n) const {ModInt ans = 1, tmp = *this;while (n) {if (n & 1) ans *= tmp;tmp *= tmp, n >>= 1;}return ans;}static constexpr int cache_limit = std::min(md, 1 << 21);static std::vector<ModInt> facs, facinvs, invs;constexpr static void _precalculation(int N) {const int l0 = facs.size();if (N > md) N = md;if (N <= l0) return;facs.resize(N), facinvs.resize(N), invs.resize(N);for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;facinvs[N - 1] = facs.back().pow(md - 2);for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];}constexpr ModInt inv() const {if (this->val_ < cache_limit) {if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);return invs[this->val_];} else {return this->pow(md - 2);}}constexpr ModInt fac() const {while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);return facs[this->val_];}constexpr ModInt facinv() const {while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);return facinvs[this->val_];}constexpr ModInt doublefac() const {lint k = (this->val_ + 1) / 2;return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()): ModInt(k).fac() * ModInt(2).pow(k);}constexpr ModInt nCr(int r) const {if (r < 0 or this->val_ < r) return ModInt(0);return this->fac() * (*this - r).facinv() * ModInt(r).facinv();}constexpr ModInt nPr(int r) const {if (r < 0 or this->val_ < r) return ModInt(0);return this->fac() * (*this - r).facinv();}static ModInt binom(int n, int r) {static long long bruteforce_times = 0;if (r < 0 or n < r) return ModInt(0);if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);r = std::min(r, n - r);ModInt ret = ModInt(r).facinv();for (int i = 0; i < r; ++i) ret *= n - i;bruteforce_times += r;return ret;}// Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)// Complexity: O(sum(ks))template <class Vec> static ModInt multinomial(const Vec &ks) {ModInt ret{1};int sum = 0;for (int k : ks) {assert(k >= 0);ret *= ModInt(k).facinv(), sum += k;}return ret * ModInt(sum).fac();}// Catalan number, C_n = binom(2n, n) / (n + 1)// C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...// https://oeis.org/A000108// Complexity: O(n)static ModInt catalan(int n) {if (n < 0) return ModInt(0);return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();}ModInt sqrt() const {if (val_ == 0) return 0;if (md == 2) return val_;if (pow((md - 1) / 2) != 1) return 0;ModInt b = 1;while (b.pow((md - 1) / 2) == 1) b += 1;int e = 0, m = md - 1;while (m % 2 == 0) m >>= 1, e++;ModInt x = pow((m - 1) / 2), y = (*this) * x * x;x *= (*this);ModInt z = b.pow(m);while (y != 1) {int j = 0;ModInt t = y;while (t != 1) j++, t *= t;z = z.pow(1LL << (e - j - 1));x *= z, z *= z, y *= z;e = j;}return ModInt(std::min(x.val_, md - x.val_));}};template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};using ModInt998244353 = ModInt<998244353>;// using mint = ModInt<998244353>;// using mint = ModInt<1000000007>;#line 5 "utilities/test/kth_power_sum.yuki665.test.cpp"using namespace std;int main() {long long n;int k;cin >> n >> k;kth_power_sum<ModInt<1000000007>> kps;cout << kps.prefix_sum(k, n) << '\n';}