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#include "fps/taylor_shift.hpp"
形式的べき級数 $f(x)$ に対して、 $f(x + c)$ を求める。 $O(N \log N)$
#pragma once #include "../fps/fps.hpp" #include "../math/binomial.hpp" #include "../modint/base.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> FormalPowerSeries<mint, convolution> taylor_shift( FormalPowerSeries<mint, convolution> f, mint a) { int d = f.deg(); Binomial<mint>::reserve(d); for (int i = 0; i < d; i++) f[i] *= Binomial<mint>::f(i); std::reverse(f.begin(), f.end()); FormalPowerSeries<mint, convolution> g(d, 1); mint pow_a = a; for (int i = 1; i < d; i++) { g[i] = pow_a * Binomial<mint>::inv_f(i); pow_a *= a; } f = (f * g).pre(d); std::reverse(f.begin(), f.end()); for (int i = 0; i < d; i++) f[i] *= Binomial<mint>::inv_f(i); return f; } } // namespace ebi
#line 2 "fps/taylor_shift.hpp" #line 2 "fps/fps.hpp" #include <algorithm> #include <cassert> #include <optional> #include <vector> #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/binomial.hpp" #include <bit> #line 6 "math/binomial.hpp" #include <ranges> #line 8 "math/binomial.hpp" #line 10 "math/binomial.hpp" namespace ebi { template <Modint mint> struct Binomial { private: static void extend(int len = -1) { int sz = (int)fact.size(); if (len < 0) len = 2 * sz; else if (len <= sz) return; else len = std::max(2 * sz, (int)std::bit_ceil(std::uint32_t(len))); len = std::min(len, mint::mod()); assert(sz <= len); fact.resize(len); inv_fact.resize(len); for (int i : std::views::iota(sz, len)) { fact[i] = fact[i - 1] * i; } inv_fact[len - 1] = fact[len - 1].inv(); for (int i : std::views::iota(sz, len) | std::views::reverse) { inv_fact[i - 1] = inv_fact[i] * i; } } public: Binomial() = default; Binomial(int n) { extend(n + 1); } static mint f(int n) { if (n >= (int)fact.size()) [[unlikely]] { extend(n + 1); } return fact[n]; } static mint inv_f(int n) { if (n >= (int)fact.size()) [[unlikely]] { extend(n + 1); } return inv_fact[n]; } static mint c(int n, int r) { if (r < 0 || n < r) return 0; return f(n) * inv_f(r) * inv_f(n - r); } static mint p(int n, int r) { if (r < 0 || n < r) return 0; return f(n) * inv_f(n - r); } static mint inv(int n) { return inv_f(n) * f(n - 1); } static void reserve(int n) { extend(n + 1); } private: static std::vector<mint> fact, inv_fact; }; template <Modint mint> std::vector<mint> Binomial<mint>::fact = std::vector<mint>(2, 1); template <Modint mint> std::vector<mint> Binomial<mint>::inv_fact = std::vector<mint>(2, 1); } // namespace ebi #line 6 "fps/taylor_shift.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> FormalPowerSeries<mint, convolution> taylor_shift( FormalPowerSeries<mint, convolution> f, mint a) { int d = f.deg(); Binomial<mint>::reserve(d); for (int i = 0; i < d; i++) f[i] *= Binomial<mint>::f(i); std::reverse(f.begin(), f.end()); FormalPowerSeries<mint, convolution> g(d, 1); mint pow_a = a; for (int i = 1; i < d; i++) { g[i] = pow_a * Binomial<mint>::inv_f(i); pow_a *= a; } f = (f * g).pre(d); std::reverse(f.begin(), f.end()); for (int i = 0; i < d; i++) f[i] *= Binomial<mint>::inv_f(i); return f; } } // namespace ebi