#include "fps/sample-point-shift.hpp"
#pragma once #include "../modulo/binomial.hpp" #include "formal-power-series.hpp" // input : y(0), y(1), ..., y(n - 1) // output : y(t), y(t + 1), ..., y(t + m - 1) // (if m is default, m = n) template <typename mint> FormalPowerSeries<mint> SamplePointShift(FormalPowerSeries<mint>& y, mint t, int m = -1) { if (m == -1) m = y.size(); long long T = t.get(); int k = (int)y.size() - 1; T %= mint::get_mod(); if (T <= k) { FormalPowerSeries<mint> ret(m); int ptr = 0; for (int64_t i = T; i <= k and ptr < m; i++) { ret[ptr++] = y[i]; } if (k + 1 < T + m) { auto suf = SamplePointShift<mint>(y, k + 1, m - ptr); for (int i = k + 1; i < T + m; i++) { ret[ptr++] = suf[i - (k + 1)]; } } return ret; } if (T + m > mint::get_mod()) { auto pref = SamplePointShift<mint>(y, T, mint::get_mod() - T); auto suf = SamplePointShift<mint>(y, 0, m - pref.size()); copy(begin(suf), end(suf), back_inserter(pref)); return pref; } FormalPowerSeries<mint> finv(k + 1, 1), d(k + 1); for (int i = 2; i <= k; i++) finv[k] *= i; finv[k] = mint(1) / finv[k]; for (int i = k; i >= 1; i--) finv[i - 1] = finv[i] * i; for (int i = 0; i <= k; i++) { d[i] = finv[i] * finv[k - i] * y[i]; if ((k - i) & 1) d[i] = -d[i]; } FormalPowerSeries<mint> h(m + k); for (int i = 0; i < m + k; i++) { h[i] = mint(1) / (T - k + i); } auto dh = d * h; FormalPowerSeries<mint> ret(m); mint cur = T; for (int i = 1; i <= k; i++) cur *= T - i; for (int i = 0; i < m; i++) { ret[i] = cur * dh[k + i]; cur *= T + i + 1; cur *= h[i]; } return ret; }
#line 2 "fps/sample-point-shift.hpp" #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 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 5 "fps/sample-point-shift.hpp" // input : y(0), y(1), ..., y(n - 1) // output : y(t), y(t + 1), ..., y(t + m - 1) // (if m is default, m = n) template <typename mint> FormalPowerSeries<mint> SamplePointShift(FormalPowerSeries<mint>& y, mint t, int m = -1) { if (m == -1) m = y.size(); long long T = t.get(); int k = (int)y.size() - 1; T %= mint::get_mod(); if (T <= k) { FormalPowerSeries<mint> ret(m); int ptr = 0; for (int64_t i = T; i <= k and ptr < m; i++) { ret[ptr++] = y[i]; } if (k + 1 < T + m) { auto suf = SamplePointShift<mint>(y, k + 1, m - ptr); for (int i = k + 1; i < T + m; i++) { ret[ptr++] = suf[i - (k + 1)]; } } return ret; } if (T + m > mint::get_mod()) { auto pref = SamplePointShift<mint>(y, T, mint::get_mod() - T); auto suf = SamplePointShift<mint>(y, 0, m - pref.size()); copy(begin(suf), end(suf), back_inserter(pref)); return pref; } FormalPowerSeries<mint> finv(k + 1, 1), d(k + 1); for (int i = 2; i <= k; i++) finv[k] *= i; finv[k] = mint(1) / finv[k]; for (int i = k; i >= 1; i--) finv[i - 1] = finv[i] * i; for (int i = 0; i <= k; i++) { d[i] = finv[i] * finv[k - i] * y[i]; if ((k - i) & 1) d[i] = -d[i]; } FormalPowerSeries<mint> h(m + k); for (int i = 0; i < m + k; i++) { h[i] = mint(1) / (T - k + i); } auto dh = d * h; FormalPowerSeries<mint> ret(m); mint cur = T; for (int i = 1; i <= k; i++) cur *= T - i; for (int i = 0; i < m; i++) { ret[i] = cur * dh[k + i]; cur *= T + i + 1; cur *= h[i]; } return ret; }