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#include "fps/product_of_one_plus_xn.hpp"
$\prod (1 + x^{a_i}) \mod x^d$ を $O(D(\log D + \log \max a))$ で求める。 $D$ は求めたい最大の次数である。中身は、$\log$ を取って $\exp$ している。
#pragma once #include <vector> #include "../fps/fps.hpp" #include "../math/mod_inv.hpp" #include "../modint/base.hpp" namespace ebi { // prod (1 + x^a_i) mod x^d template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> FormalPowerSeries<mint, convolution> product_of_one_plus_xn(std::vector<int> a, int d) { using FPS = FormalPowerSeries<mint, convolution>; std::vector<int> cnt(d, 0); for (auto x : a) if (x < d) cnt[x]++; FPS log_f(d); for (int x = 1; x < d; x++) { for (int i = 1; x * i < d; i++) { if (i & 1) log_f[x * i] += mint(cnt[x]) * inv<mint>(i); else log_f[x * i] -= mint(cnt[x]) * inv<mint>(i); } } mint ret = mint(2).pow(cnt[0]); auto f = log_f.exp(d); for (auto &x : f) x *= ret; return f; } } // namespace ebi
#line 2 "fps/product_of_one_plus_xn.hpp" #include <vector> #line 2 "fps/fps.hpp" #include <algorithm> #include <cassert> #include <optional> #line 7 "fps/fps.hpp" #line 2 "modint/base.hpp" #include <concepts> #include <iostream> #include <utility> namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 9 "fps/fps.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> struct FormalPowerSeries : std::vector<mint> { private: using std::vector<mint>::vector; using std::vector<mint>::vector::operator=; using FPS = FormalPowerSeries; public: FormalPowerSeries(const std::vector<mint> &a) { *this = a; } FPS operator+(const FPS &rhs) const noexcept { return FPS(*this) += rhs; } FPS operator-(const FPS &rhs) const noexcept { return FPS(*this) -= rhs; } FPS operator*(const FPS &rhs) const noexcept { return FPS(*this) *= rhs; } FPS operator/(const FPS &rhs) const noexcept { return FPS(*this) /= rhs; } FPS operator%(const FPS &rhs) const noexcept { return FPS(*this) %= rhs; } FPS operator+(const mint &rhs) const noexcept { return FPS(*this) += rhs; } FPS operator-(const mint &rhs) const noexcept { return FPS(*this) -= rhs; } FPS operator*(const mint &rhs) const noexcept { return FPS(*this) *= rhs; } FPS operator/(const mint &rhs) const noexcept { return FPS(*this) /= rhs; } FPS &operator+=(const FPS &rhs) noexcept { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) { (*this)[i] += rhs[i]; } return *this; } FPS &operator-=(const FPS &rhs) noexcept { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) { (*this)[i] -= rhs[i]; } return *this; } FPS &operator*=(const FPS &rhs) noexcept { *this = convolution(*this, rhs); return *this; } FPS &operator/=(const FPS &rhs) noexcept { int n = deg() - 1; int m = rhs.deg() - 1; if (n < m) { *this = {}; return *this; } *this = (*this).rev() * rhs.rev().inv(n - m + 1); (*this).resize(n - m + 1); std::reverse((*this).begin(), (*this).end()); return *this; } FPS &operator%=(const FPS &rhs) noexcept { *this -= *this / rhs * rhs; shrink(); return *this; } FPS &operator+=(const mint &rhs) noexcept { if (this->empty()) this->resize(1); (*this)[0] += rhs; return *this; } FPS &operator-=(const mint &rhs) noexcept { if (this->empty()) this->resize(1); (*this)[0] -= rhs; return *this; } FPS &operator*=(const mint &rhs) noexcept { for (int i = 0; i < deg(); ++i) { (*this)[i] *= rhs; } return *this; } FPS &operator/=(const mint &rhs) noexcept { mint inv_rhs = rhs.inv(); for (int i = 0; i < deg(); ++i) { (*this)[i] *= inv_rhs; } return *this; } FPS operator>>(int d) const { if (deg() <= d) return {}; FPS f = *this; f.erase(f.begin(), f.begin() + d); return f; } FPS operator<<(int d) const { FPS f = *this; f.insert(f.begin(), d, 0); return f; } FPS operator-() const { FPS g(this->size()); for (int i = 0; i < (int)this->size(); i++) g[i] = -(*this)[i]; return g; } FPS pre(int sz) const { return FPS(this->begin(), this->begin() + std::min(deg(), sz)); } FPS rev() const { auto f = *this; std::reverse(f.begin(), f.end()); return f; } FPS differential() const { int n = deg(); FPS g(std::max(0, n - 1)); for (int i = 0; i < n - 1; i++) { g[i] = (*this)[i + 1] * (i + 1); } return g; } FPS integral() const { int n = deg(); FPS g(n + 1); g[0] = 0; if (n > 0) g[1] = 1; auto mod = mint::mod(); for (int i = 2; i <= n; i++) g[i] = (-g[mod % i]) * (mod / i); for (int i = 0; i < n; i++) g[i + 1] *= (*this)[i]; return g; } FPS inv(int d = -1) const { int n = 1; if (d < 0) d = deg(); FPS g(n); g[0] = (*this)[0].inv(); while (n < d) { n <<= 1; g = (g * 2 - g * g * this->pre(n)).pre(n); } g.resize(d); return g; } FPS log(int d = -1) const { assert((*this)[0].val() == 1); if (d < 0) d = deg(); return ((*this).differential() * (*this).inv(d)).pre(d - 1).integral(); } FPS exp(int d = -1) const { assert((*this)[0].val() == 0); int n = 1; if (d < 0) d = deg(); FPS g(n); g[0] = 1; while (n < d) { n <<= 1; g = (g * (this->pre(n) - g.log(n) + 1)).pre(n); } g.resize(d); return g; } FPS pow(int64_t k, int d = -1) const { const int n = deg(); if (d < 0) d = n; if (k == 0) { FPS f(d); if (d > 0) f[0] = 1; return f; } for (int i = 0; i < n; i++) { if ((*this)[i] != 0) { mint rev = (*this)[i].inv(); FPS f = (((*this * rev) >> i).log(d) * k).exp(d); f *= (*this)[i].pow(k); f = (f << (i * k)).pre(d); if (f.deg() < d) f.resize(d); return f; } if (i + 1 >= (d + k - 1) / k) break; } return FPS(d); } int deg() const { return (*this).size(); } void shrink() { while ((!this->empty()) && this->back() == 0) this->pop_back(); } int count_terms() const { int c = 0; for (int i = 0; i < deg(); i++) { if ((*this)[i] != 0) c++; } return c; } std::optional<FPS> sqrt(int d = -1) const; static FPS exp_x(int n) { FPS f(n); mint fact = 1; for (int i = 1; i < n; i++) fact *= i; f[n - 1] = fact.inv(); for (int i = n - 1; i >= 0; i--) f[i - 1] = f[i] * i; return f; } }; } // namespace ebi #line 2 "math/mod_inv.hpp" #line 5 "math/mod_inv.hpp" #line 7 "math/mod_inv.hpp" namespace ebi { template <Modint mint> mint inv(int n) { static const int mod = mint::mod(); static std::vector<mint> dat = {0, 1}; assert(0 <= n); if (n >= mod) n -= mod; while (int(dat.size()) <= n) { int num = dat.size(); int q = (mod + num - 1) / num; dat.emplace_back(dat[num * q - mod] * mint(q)); } return dat[n]; } } // namespace ebi #line 8 "fps/product_of_one_plus_xn.hpp" namespace ebi { // prod (1 + x^a_i) mod x^d template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> FormalPowerSeries<mint, convolution> product_of_one_plus_xn(std::vector<int> a, int d) { using FPS = FormalPowerSeries<mint, convolution>; std::vector<int> cnt(d, 0); for (auto x : a) if (x < d) cnt[x]++; FPS log_f(d); for (int x = 1; x < d; x++) { for (int i = 1; x * i < d; i++) { if (i & 1) log_f[x * i] += mint(cnt[x]) * inv<mint>(i); else log_f[x * i] -= mint(cnt[x]) * inv<mint>(i); } } mint ret = mint(2).pow(cnt[0]); auto f = log_f.exp(d); for (auto &x : f) x *= ret; return f; } } // namespace ebi