平行移動
(fps/taylor-shift.hpp)

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• Last update: 2026-06-08 17:59:24+09:00
• Include: #include "fps/taylor-shift.hpp"

fps-平行移動

$N$次の多項式 $f(x)$ に対して $g(x) = f(x+a)$ を満たす $g(x)$ を $\mathrm{O}(N \log N)$ で計算するライブラリ。

概要

$N$ 次の多項式 $f(x)$ に対して $g(x) = f(x+a)$ となる $g(x)$ を求める。これは、

\[f(x+a) = \sum_{n=0}^N f_n(x+a)^n\] \[=\sum_{i=0}^Nx^i\sum_{n=i}^N(a^{n-i}f_n\ _nC_i)\]

$m=N-n,j=N-i$ とおいて

\[=\sum_{j=0}^N\frac{x^j}{(N-j)!}\sum_{m=0}^j\frac{a^{j-m}}{(j-m)!}\left((N-m)!f_{N-m}\right)\]

と変形できるので畳み込みに帰着される。

使い方

• TaylorShift(f, a, C): 多項式 $f$ について $f(x+a)$ を f.size() 次未満の項まで求める。 $C$ には Binomial 構造体を渡す。

Depends on

• 多項式/形式的冪級数ライブラリ (fps/formal-power-series.hpp)
• modulo/binomial.hpp

Required by

• 有名な数列 (fps/fps-famous-series.hpp)
• fps/fualhuber.hpp
• 部分分数分解(分母が1次式の積で表せる場合) (fps/partial-fraction-decomposition.hpp)
• 集合冪級数の合成 (set-function/polynomial-composite-set-power-series.hpp)

Verified with

• verify/verify-unit-test/partial-fraction-decomposition.test.cpp
• verify/verify-yosupo-fps/yosupo-stirling-1st-row.test.cpp
• verify/verify-yosupo-fps/yosupo-stirling-1st.test.cpp
• verify/verify-yosupo-fps/yosupo-stirling-2nd-row.test.cpp
• verify/verify-yosupo-fps/yosupo-stirling-2nd.test.cpp
• verify/verify-yosupo-fps/yosupo-taylor-shift.test.cpp
• verify/verify-yosupo-math/yosupo-polynomial-composite-set-power-series.test.cpp
• verify/verify-yuki/yuki-1504.test.cpp
• verify/verify-yuki/yuki-1510.test.cpp
• verify/verify-yuki/yuki-1875.test.cpp
• verify/verify-yuki/yuki-2580.test.cpp

Code

#pragma once
#include "./formal-power-series.hpp"
#include "../modulo/binomial.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
 */

#line 2 "fps/formal-power-series.hpp"

#include <algorithm>
#include <cassert>
#include <cstdint>
#include <iterator>
#include <vector>
using namespace std;

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;

template <int N>
struct FPSBackendPriority : FPSBackendPriority<N - 1> {};
template <>
struct FPSBackendPriority<0> {};

template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
  fps_set_fft_impl((FormalPowerSeries<mint>*)nullptr, FPSBackendPriority<1>{});
}

template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FPS& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  return fps_multiply_impl(*this, r, FPSBackendPriority<1>{});
}

template <typename mint>
void FormalPowerSeries<mint>::ntt() {
  fps_ntt_impl(*this, FPSBackendPriority<1>{});
}

template <typename mint>
void FormalPowerSeries<mint>::intt() {
  fps_intt_impl(*this, FPSBackendPriority<1>{});
}

template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
  fps_ntt_doubling_impl(*this, FPSBackendPriority<1>{});
}

template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
  return fps_ntt_pr_impl((FormalPowerSeries<mint>*)nullptr,
                         FPSBackendPriority<1>{});
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  return fps_inv_impl(*this, deg, FPSBackendPriority<1>{});
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  return fps_exp_impl(*this, deg, FPSBackendPriority<1>{});
}

/**
 * @brief 多項式/形式的冪級数ライブラリ
 */
#line 2 "modulo/binomial.hpp"

#line 4 "modulo/binomial.hpp"
#include <type_traits>
#line 6 "modulo/binomial.hpp"
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
 */

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