#include "fps/partial-fraction-decomposition.hpp"
#pragma once #include "formal-power-series.hpp" #include "taylor-shift.hpp" #include "../modulo/binomial.hpp" // 新しい方 // 入力 : f/(x-b_1)(x-b_2)...(x-b_n), f は n-1 次, b_i は distinct // 出力 : a_1/(x-b_1) + ... + a_n/(x-b_n) template <typename mint> vector<mint> PartialFractionDecomposition(const FormalPowerSeries<mint>& f, const vector<mint>& b) { using fps = FormalPowerSeries<mint>; int N = b.size(); assert((int)f.size() <= N); int B = 1; while (B < N) B *= 2; vector<fps> mod(2 * B, fps{1}); for (int i = 0; i < N; i++) mod[B + i] = fps{-b[i], 1}; for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1]; vector<mint> ans(N); auto dfs = [&](auto rc, int i, int l, int r, fps g, fps h) -> void { if (N <= l) return; if (l + 1 == r) { ans[l] = g.eval(0) / h.eval(0); return; } int m = (l + r) / 2; rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0], h * mod[i * 2 + 1] % mod[i * 2]); rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1], h * mod[i * 2] % mod[i * 2 + 1]); }; dfs(dfs, 1, 0, B, f, fps{1}); return ans; } // 古い方 template <typename mint> vector<pair<mint, vector<mint>>> PartialFractionDecomposition( FormalPowerSeries<mint> numer, vector<pair<mint, int>> denom) { using fps = FormalPowerSeries<mint>; if (denom.empty()) return {}; sort(begin(denom), end(denom), [](auto p1, auto p2) { return p1.second < p2.second; }); Binomial<mint> C(denom[0].second + 1); int s = 1; while (s < (int)denom.size()) s *= 2; vector<fps> fs(2 * s); for (int i = 0; i < s; i++) { if ((int)denom.size() <= i) { fs[s + i] = fps{1}; continue; } auto [m, d] = denom[i]; fps f(denom[i].second + 1); mint buf = 1; for (int j = d; j >= 0; j--) { f[j] = buf * C(d, j); buf *= m; } fs[s + i] = f; } for (int i = s - 1; i; i--) { fs[i] = fs[2 * i + 0] * fs[2 * i + 1]; } vector<fps> F(2 * s); vector<fps> G(2 * s); F[1] = numer % fs[1]; G[1] = fps{1}; for (int i = 1; i < s; i++) { F[i * 2 + 0] = F[i] % fs[i * 2 + 0]; F[i * 2 + 1] = F[i] % fs[i * 2 + 1]; G[i * 2 + 0] = G[i] * fs[i * 2 + 1] % fs[i * 2 + 0]; G[i * 2 + 1] = G[i] * fs[i * 2 + 0] % fs[i * 2 + 1]; } vector<pair<mint, vector<mint>>> res; for (int i = s; i - s < (int)denom.size(); i++) { auto [m, d] = denom[i - s]; F[i] = TaylorShift<mint>(F[i], -m, C); G[i] = TaylorShift<mint>(G[i], -m, C); fps f = (F[i] * G[i].inv()).pre(d); if ((int)f.size() < d) f.resize(d); f = f.rev(); res.emplace_back(m, vector<mint>{begin(f), end(f)}); } return res; } /** * @brief 部分分数分解(分母が1次式の積で表せる場合) */
#line 2 "fps/partial-fraction-decomposition.hpp" #line 2 "fps/formal-power-series.hpp" template <typename mint> struct FormalPowerSeries : vector<mint> { using vector<mint>::vector; using FPS = FormalPowerSeries; FPS &operator+=(const FPS &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i]; return *this; } FPS &operator+=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] += r; return *this; } FPS &operator-=(const FPS &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i]; return *this; } FPS &operator-=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] -= r; return *this; } FPS &operator*=(const mint &v) { for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v; return *this; } FPS &operator/=(const FPS &r) { if (this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; if ((int)r.size() <= 64) { FPS f(*this), g(r); g.shrink(); mint coeff = g.back().inverse(); for (auto &x : g) x *= coeff; int deg = (int)f.size() - (int)g.size() + 1; int gs = g.size(); FPS quo(deg); for (int i = deg - 1; i >= 0; i--) { quo[i] = f[i + gs - 1]; for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j]; } *this = quo * coeff; this->resize(n, mint(0)); return *this; } return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev(); } FPS &operator%=(const FPS &r) { *this -= *this / r * r; shrink(); return *this; } FPS operator+(const FPS &r) const { return FPS(*this) += r; } FPS operator+(const mint &v) const { return FPS(*this) += v; } FPS operator-(const FPS &r) const { return FPS(*this) -= r; } FPS operator-(const mint &v) const { return FPS(*this) -= v; } FPS operator*(const FPS &r) const { return FPS(*this) *= r; } FPS operator*(const mint &v) const { return FPS(*this) *= v; } FPS operator/(const FPS &r) const { return FPS(*this) /= r; } FPS operator%(const FPS &r) const { return FPS(*this) %= r; } FPS operator-() const { FPS ret(this->size()); for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i]; return ret; } void shrink() { while (this->size() && this->back() == mint(0)) this->pop_back(); } FPS rev() const { FPS ret(*this); reverse(begin(ret), end(ret)); return ret; } FPS dot(FPS r) const { FPS ret(min(this->size(), r.size())); for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする FPS pre(int sz) const { FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz)); if ((int)ret.size() < sz) ret.resize(sz); return ret; } FPS operator>>(int sz) const { if ((int)this->size() <= sz) return {}; FPS ret(*this); ret.erase(ret.begin(), ret.begin() + sz); return ret; } FPS operator<<(int sz) const { FPS ret(*this); ret.insert(ret.begin(), sz, mint(0)); return ret; } FPS diff() const { const int n = (int)this->size(); FPS ret(max(0, n - 1)); mint one(1), coeff(1); for (int i = 1; i < n; i++) { ret[i - 1] = (*this)[i] * coeff; coeff += one; } return ret; } FPS integral() const { const int n = (int)this->size(); FPS ret(n + 1); ret[0] = mint(0); if (n > 0) ret[1] = mint(1); auto mod = mint::get_mod(); for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i); for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i]; return ret; } mint eval(mint x) const { mint r = 0, w = 1; for (auto &v : *this) r += w * v, w *= x; return r; } FPS log(int deg = -1) const { assert(!(*this).empty() && (*this)[0] == mint(1)); if (deg == -1) deg = (int)this->size(); return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } FPS pow(int64_t k, int deg = -1) const { const int n = (int)this->size(); if (deg == -1) deg = n; if (k == 0) { FPS ret(deg); if (deg) ret[0] = 1; return ret; } for (int i = 0; i < n; i++) { if ((*this)[i] != mint(0)) { mint rev = mint(1) / (*this)[i]; FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg); ret *= (*this)[i].pow(k); ret = (ret << (i * k)).pre(deg); if ((int)ret.size() < deg) ret.resize(deg, mint(0)); return ret; } if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0)); } return FPS(deg, mint(0)); } static void *ntt_ptr; static void set_fft(); FPS &operator*=(const FPS &r); void ntt(); void intt(); void ntt_doubling(); static int ntt_pr(); FPS inv(int deg = -1) const; FPS exp(int deg = -1) const; }; template <typename mint> void *FormalPowerSeries<mint>::ntt_ptr = nullptr; /** * @brief 多項式/形式的冪級数ライブラリ * @docs docs/fps/formal-power-series.md */ #line 2 "modulo/binomial.hpp" #include <cassert> #include <type_traits> #include <vector> using namespace std; // コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」 // を入れると倍速くらいになる // mod を超えて前計算して 0 割りを踏むバグは対策済み template <typename T> struct Binomial { vector<T> f, g, h; Binomial(int MAX = 0) { assert(T::get_mod() != 0 && "Binomial<mint>()"); f.resize(1, T{1}); g.resize(1, T{1}); h.resize(1, T{1}); if (MAX > 0) extend(MAX + 1); } void extend(int m = -1) { int n = f.size(); if (m == -1) m = n * 2; m = min<int>(m, T::get_mod()); if (n >= m) return; f.resize(m); g.resize(m); h.resize(m); for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i); g[m - 1] = f[m - 1].inverse(); h[m - 1] = g[m - 1] * f[m - 2]; for (int i = m - 2; i >= n; i--) { g[i] = g[i + 1] * T(i + 1); h[i] = g[i] * f[i - 1]; } } T fac(int i) { if (i < 0) return T(0); while (i >= (int)f.size()) extend(); return f[i]; } T finv(int i) { if (i < 0) return T(0); while (i >= (int)g.size()) extend(); return g[i]; } T inv(int i) { if (i < 0) return -inv(-i); while (i >= (int)h.size()) extend(); return h[i]; } T C(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); return fac(n) * finv(n - r) * finv(r); } inline T operator()(int n, int r) { return C(n, r); } template <typename I> T multinomial(const vector<I>& r) { static_assert(is_integral<I>::value == true); int n = 0; for (auto& x : r) { if (x < 0) return T(0); n += x; } T res = fac(n); for (auto& x : r) res *= finv(x); return res; } template <typename I> T operator()(const vector<I>& r) { return multinomial(r); } T C_naive(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); T ret = T(1); r = min(r, n - r); for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--); return ret; } T P(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); return fac(n) * finv(n - r); } // [x^r] 1 / (1-x)^n T H(int n, int r) { if (n < 0 || r < 0) return T(0); return r == 0 ? 1 : C(n + r - 1, r); } }; #line 4 "fps/taylor-shift.hpp" // calculate F(x + a) template <typename mint> FormalPowerSeries<mint> TaylorShift(FormalPowerSeries<mint> f, mint a, Binomial<mint>& C) { using fps = FormalPowerSeries<mint>; int N = f.size(); for (int i = 0; i < N; i++) f[i] *= C.fac(i); reverse(begin(f), end(f)); fps g(N, mint(1)); for (int i = 1; i < N; i++) g[i] = g[i - 1] * a * C.inv(i); f = (f * g).pre(N); reverse(begin(f), end(f)); for (int i = 0; i < N; i++) f[i] *= C.finv(i); return f; } /** * @brief 平行移動 * @docs docs/fps/fps-taylor-shift.md */ #line 6 "fps/partial-fraction-decomposition.hpp" // 新しい方 // 入力 : f/(x-b_1)(x-b_2)...(x-b_n), f は n-1 次, b_i は distinct // 出力 : a_1/(x-b_1) + ... + a_n/(x-b_n) template <typename mint> vector<mint> PartialFractionDecomposition(const FormalPowerSeries<mint>& f, const vector<mint>& b) { using fps = FormalPowerSeries<mint>; int N = b.size(); assert((int)f.size() <= N); int B = 1; while (B < N) B *= 2; vector<fps> mod(2 * B, fps{1}); for (int i = 0; i < N; i++) mod[B + i] = fps{-b[i], 1}; for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1]; vector<mint> ans(N); auto dfs = [&](auto rc, int i, int l, int r, fps g, fps h) -> void { if (N <= l) return; if (l + 1 == r) { ans[l] = g.eval(0) / h.eval(0); return; } int m = (l + r) / 2; rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0], h * mod[i * 2 + 1] % mod[i * 2]); rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1], h * mod[i * 2] % mod[i * 2 + 1]); }; dfs(dfs, 1, 0, B, f, fps{1}); return ans; } // 古い方 template <typename mint> vector<pair<mint, vector<mint>>> PartialFractionDecomposition( FormalPowerSeries<mint> numer, vector<pair<mint, int>> denom) { using fps = FormalPowerSeries<mint>; if (denom.empty()) return {}; sort(begin(denom), end(denom), [](auto p1, auto p2) { return p1.second < p2.second; }); Binomial<mint> C(denom[0].second + 1); int s = 1; while (s < (int)denom.size()) s *= 2; vector<fps> fs(2 * s); for (int i = 0; i < s; i++) { if ((int)denom.size() <= i) { fs[s + i] = fps{1}; continue; } auto [m, d] = denom[i]; fps f(denom[i].second + 1); mint buf = 1; for (int j = d; j >= 0; j--) { f[j] = buf * C(d, j); buf *= m; } fs[s + i] = f; } for (int i = s - 1; i; i--) { fs[i] = fs[2 * i + 0] * fs[2 * i + 1]; } vector<fps> F(2 * s); vector<fps> G(2 * s); F[1] = numer % fs[1]; G[1] = fps{1}; for (int i = 1; i < s; i++) { F[i * 2 + 0] = F[i] % fs[i * 2 + 0]; F[i * 2 + 1] = F[i] % fs[i * 2 + 1]; G[i * 2 + 0] = G[i] * fs[i * 2 + 1] % fs[i * 2 + 0]; G[i * 2 + 1] = G[i] * fs[i * 2 + 0] % fs[i * 2 + 1]; } vector<pair<mint, vector<mint>>> res; for (int i = s; i - s < (int)denom.size(); i++) { auto [m, d] = denom[i - s]; F[i] = TaylorShift<mint>(F[i], -m, C); G[i] = TaylorShift<mint>(G[i], -m, C); fps f = (F[i] * G[i].inv()).pre(d); if ((int)f.size() < d) f.resize(d); f = f.rev(); res.emplace_back(m, vector<mint>{begin(f), end(f)}); } return res; } /** * @brief 部分分数分解(分母が1次式の積で表せる場合) */