P-recursive
(fps/p-recursive.hpp)

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• Last update: 2026-07-28 20:47:27+09:00
• Include: #include "fps/p-recursive.hpp"

P-recursive 関連のライブラリ．

P-recursive について

数列 $(a_n)_{n=0}^{\infty}$ が P-recursive であるとは，ある非負整数 $r$ および多項式 $p_0,p_1,\dots,p_r$（一つ以上は $0$ でない）が存在して，任意の $n\geq 0$ に対して

\[p_0(n)a_n+p_1(n)a_{n+1}+\cdots+p_r(n)a_{n+r}=0\]

を満たすことをいう．

使い方

有限体上の数列 $a=(a_0,a_1,\dots,a_{N-1})$ を入力とする．推定された漸化式の次数を $r$，$D=\max(1,d)$，長さ $n$ の畳み込みの時間計算量を $M(n)$ とする．

• FindPRecursive(a, d)：$\sum_{j=0}^r f_j(i)a_{i+j}=0$ を満たす次数 $d$ 以下の多項式を $(f_r,f_{r-1},\dots,f_0)$ の順で推定する．解が得られなければ空列を返す．$O(((r+1)(d+1))^3)$ 時間．
• PRecursiveTerm(a, k, d)：係数多項式の次数を $d$ として漸化式を推定し，$a_k$ を求める．$O(((r+1)(d+1))^3+r^3\sqrt{kD}+r^2M(\sqrt{kD}))$ 時間．
• EnumeratePRecursive(a, n, d)：係数多項式の次数を $d$ として漸化式を推定し，$a_0,a_1,\dots,a_{n-1}$ を求める．$n\gt N$ のとき $O(((r+1)(d+1))^3+(n-N)r(d+1))$ 時間．
• PRecursiveTerm(a, k)：$d$ を小さい方から探索し，入力の末尾 1 項で検算してから $a_k$ を求める．計算量は試した各 $d$ に対する上記の計算量の総和．

アルゴリズム

漸化式の推定

各 $f_j$ を

\[f_j(i)=\sum_{l=0}^d c_{j,l}i^l\]

と置く．既知の各項について

\[\sum_{j=0}^r\sum_{l=0}^d c_{j,l}i^la_{i+j}=0\]

を立てると，未知数 $c_{j,l}$ に関する斉次線形方程式になる．非自明な解を一つ求めれば漸化式を得る．未知数は $(r+1)(d+1)$ 個であり，Gauss の消去法によりおよそ $O(((r+1)(d+1))^3)$ 時間．

第 $k$ 項

$f_r(i)\neq 0$ とする．状態ベクトル

\[v_i=(a_{i+r-1},a_{i+r-2},\dots,a_i)^\mathsf{T}\]

に対し，漸化式を

\[v_{i+1}=\frac{1}{f_r(i)}M(i)v_i\]

と書ける．$M(i)$ は，最上段が

\[(-f_{r-1}(i),-f_{r-2}(i),\dots,-f_0(i))\]

で，直下の対角成分が全て $f_r(i)$ の多項式行列である．したがって $a_k$ は

\[\frac{M(k-r)M(k-r-1)\cdots M(0)} {\prod_{i=0}^{k-r}f_r(i)}v_0\]

の先頭成分として求められる．

$D=\max(1,d)$ とする．多項式行列の積は baby-step giant-step で計算する．長さをおよそ $\sqrt{k/D}$ とした区間ごとの積を用意し，各成分を多項式とみなして評価点シフトを行う．これにより，$k$ 個の行列を直接掛けず，$k$ に対して平方根程度の個数の行列積と多項式の評価点シフトへ帰着する．

推定された漸化式は証明ではない．構成に使わなかった項でも検算する必要がある．また，$a_k$ までの計算に現れる最高次係数 $f_r(i)$ は全て $0$ でない必要がある．

参考

• Nyaan’s Library：P-recursive の高速計算

Depends on

• fps/formal-power-series.hpp
• 評価点シフト (fps/sampling-points-shift.hpp)
• matrix/matrix.hpp
• 多項式行列の prefix product (matrix/polynomial-matrix-prefix-product.hpp)
• 階乗, 二項係数 (modint/factorial.hpp)

Verified with

• verify/fps/LC_factorial.p_recursive.test.cpp

Code

#pragma once
#include "fps/formal-power-series.hpp"
#include "matrix/polynomial-matrix-prefix-product.hpp"

namespace p_recursive {

template <class T>
vector<T> NullVector(vector<vector<T>> a) {
  int h = a.size(), w = a[0].size(), rank = 0;
  vector<int> pivot;
  for (int j = 0; j < w && rank < h; j++) {
    int k = rank;
    while (k < h && a[k][j] == T{}) k++;
    if (k == h) continue;
    swap(a[rank], a[k]);
    T iv = T(1) / a[rank][j];
    for (int l = j; l < w; l++) a[rank][l] *= iv;
    for (int i = 0; i < h; i++) {
      if (i == rank || a[i][j] == T{}) continue;
      T c = a[i][j];
      for (int l = j; l < w; l++) a[i][l] -= c * a[rank][l];
    }
    pivot.push_back(j);
    rank++;
  }
  if (rank == w) return {};
  vector<bool> is_pivot(w, false);
  for (int j : pivot) is_pivot[j] = true;
  int free = 0;
  while (is_pivot[free]) free++;
  vector<T> x(w);
  x[free] = 1;
  for (int i = 0; i < rank; i++) x[pivot[i]] = -a[i][free];
  return x;
}

}  // namespace p_recursive

// sum[j=0...r] f[j](i) a[i+j] = 0 を満たす f を降順に返す
template <class mint>
vector<FormalPowerSeries<mint>> FindPRecursive(const vector<mint>& a, int d) {
  using fps = FormalPowerSeries<mint>;
  assert(d >= 0);
  int n = a.size();
  int r = (n + 2) / (d + 2) - 1;
  if (r <= 0) return {};
  int m = (r + 1) * (d + 1);
  vector<vector<mint>> mat(m - 1, vector<mint>(m));
  for (int i = 0; i < m - 1; i++) {
    for (int j = 0; j <= r; j++) {
      mint x = 1;
      for (int l = 0; l <= d; l++) {
        mat[i][(d + 1) * j + l] = x * a[i + j];
        x *= i + j;
      }
    }
  }
  auto c = p_recursive::NullVector(mat);
  if (c.empty()) return {};
  while ((int)c.size() > d + 1 &&
         all_of(c.end() - d - 1, c.end(), [](mint x) { return x == mint(0); }))
    c.erase(c.end() - d - 1, c.end());
  vector<fps> ret;
  for (int i = 0, j = 0; i < (int)c.size(); i += d + 1, j++) {
    fps f{1}, base{mint(j), mint(1)}, sum;
    for (int l = 0; l <= d; l++) {
      sum += f * c[i + l];
      f *= base;
    }
    sum.shrink();
    ret.push_back(sum);
  }
  reverse(ret.begin(), ret.end());
  return ret;
}

template <class mint>
mint PRecursiveTerm(const vector<mint>& a, long long k, int d) {
  if (k < (long long)a.size()) return a[k];
  if (all_of(a.begin(), a.end(), [](mint x) { return x == mint(0); })) return 0;
  auto f = FindPRecursive(a, d);
  assert(f.size() >= 2);
  int r = f.size() - 1;
  Matrix<FormalPowerSeries<mint>> m(r), denom(1);
  for (int i = 0; i < r; i++) m.set(0, i, -f[i + 1]);
  for (int i = 1; i < r; i++) m.set(i, i - 1, f[0]);
  denom.set(0, 0, f[0]);
  Matrix<mint> init(r, 1);
  for (int i = 0; i < r; i++) init.set(i, 0, a[r - 1 - i]);
  mint ret = (PolynomialMatrixPrefixProduct(m, k - r + 1) * init).get(0, 0);
  ret /= PolynomialMatrixPrefixProduct(denom, k - r + 1).get(0, 0);
  return ret;
}

template <class mint>
vector<mint> EnumeratePRecursive(const vector<mint>& a, long long n, int d) {
  assert(n >= 0);
  if (n <= (long long)a.size()) return vector<mint>(a.begin(), a.begin() + n);
  if (all_of(a.begin(), a.end(), [](mint x) { return x == mint(0); })) return vector<mint>(n);
  auto f = FindPRecursive(a, d);
  if (f.size() < 2) return {};
  int r = f.size() - 1;
  reverse(f.begin(), f.end());
  vector<mint> ret(a);
  ret.resize(n);
  for (long long k = a.size(); k < n; k++) {
    long long i = k - r;
    mint sum = 0;
    for (int j = 0; j < r; j++) sum += ret[i + j] * f[j].eval(mint(i));
    mint c = f[r].eval(mint(i));
    assert(c != mint(0));
    ret[k] = -sum / c;
  }
  return ret;
}

template <class mint>
mint PRecursiveTerm(const vector<mint>& a, long long k) {
  if (k < (long long)a.size()) return a[k];
  if (all_of(a.begin(), a.end(), [](mint x) { return x == mint(0); })) return 0;
  assert(a.size() >= 2);
  vector<mint> b(a.begin(), a.end() - 1);
  int n = a.size() - 1;
  for (int d = 0; (n + 2) / (d + 2) > 1; d++) {
    auto f = FindPRecursive(b, d);
    if (f.size() < 2) continue;
    auto c = EnumeratePRecursive(b, n + 1, d);
    if (!c.empty() && c.back() == a.back()) return PRecursiveTerm(a, k, d);
  }
  assert(false);
  return 0;
}

/**
 * @brief P-recursive
 * @docs docs/fps/p-recursive.md
 */

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

template <class mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;
  FormalPowerSeries(const vector<mint>& r) : vector<mint>(r) {}
  FormalPowerSeries(vector<mint>&& r) : vector<mint>(std::move(r)) {}
  FPS& operator=(const vector<mint>& r) {
    vector<mint>::operator=(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 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.at(g.size() - 1).inv();
      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;
  }
  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }
  FPS operator>>=(int sz) {
    assert(sz >= 0);
    if ((int)this->size() <= sz)
      this->clear();
    else
      this->erase(this->begin(), this->begin() + sz);
    return *this;
  }
  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) {
    assert(sz >= 0);
    this->insert(this->begin(), sz, mint(0));
    return *this;
  }
  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)[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_ntt();
  FPS& operator*=(const FPS& r);
  FPS middle_product(const FPS& r) const;
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_root();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void* FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 2 "modint/factorial.hpp"

template <class mint>
struct Factorial {
  static void reserve(int n) {
    inv(n);
    fact(n);
    fact_inv(n);
  }
  static mint inv(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({0, 1});
    assert(n != 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({0, 1});
    }
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size(), q = (mod + k - 1) / k;
      buf.push_back(q * buf[k * q - mod]);
    }
    return buf[n];
  }
  static mint fact(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({1, 1});
    assert(n >= 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({1, 1});
    }
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size();
      buf.push_back(buf.back() * k);
    }
    return buf[n];
  }
  static mint fact_inv(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({1, 1});
    assert(n >= 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({1, 1});
    }
    if ((int)buf.size() <= n) inv(n);
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size();
      buf.push_back(buf.back() * inv(k));
    }
    return buf[n];
  }
  static mint binom(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact(n) * fact_inv(r) * fact_inv(n - r);
  }
  static mint binom_naive(int n, int r) {
    if (r < 0 || r > n) return 0;
    mint res = fact_inv(r);
    for (int i = 0; i < r; i++) res *= n - i;
    return res;
  }
  static mint multinom(const vector<int>& r) {
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return 0;
      n += x;
    }
    mint res = fact(n);
    for (auto& x : r) res *= fact_inv(x);
    return res;
  }
  static mint P(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact(n) * fact_inv(n - r);
  }
  // partition n items to r groups (allow empty group)
  static mint H(int n, int r) {
    if (n < 0 || r < 0) return 0;
    return r == 0 ? 1 : binom(n + r - 1, r);
  }
};
/**
 * @brief 階乗, 二項係数
 */
#line 4 "fps/sampling-points-shift.hpp"

// f(0),f(1),...,f(n-1) -> f(c),...,f(c+m-1)
template <class mint>
vector<mint> SamplingPointsShift(const vector<mint>& f, mint c, int m) {
  using fps = FormalPowerSeries<mint>;
  using fact = Factorial<mint>;
  int n = f.size();
  fact::reserve(m);
  fps f1(n), ei(n);
  for (int i = 0; i < n; i++) f1[i] = f[i] * fact::fact_inv(i);
  for (int i = 0; i < n; i++) ei[i] = fact::fact_inv(i) * (i % 2 ? -1 : 1);
  f1 *= ei;
  for (int i = n; i < f1.size(); i++) f1[i] = 0;
  for (int i = 0; i < n; i++) f1[i] *= fact::fact(i);
  fps g(n, 1);
  for (int i = 1; i < n; i++) g[i] = g[i - 1] * (c + 1 - i) * fact::inv(i);
  g = g.middle_product(f1);
  for (int i = 0; i < n; i++) g[i] *= fact::fact_inv(i);
  fps e(m);
  for (int i = 0; i < m; i++) e[i] = fact::fact_inv(i);
  g *= e;
  g.resize(m);
  for (int i = 0; i < m; i++) g[i] *= fact::fact(i);
  return g;
}
/**
 * @brief 評価点シフト
 * @docs docs/fps/sampling-points-shift.md
 */
#line 2 "matrix/matrix.hpp"

template <class T>
struct Matrix {
  int h, w;
  vector<T> a;
  Matrix() {}
  Matrix(int n) : h(n), w(n), a(n * n, T{}) {}
  Matrix(int h_, int w_) : h(h_), w(w_), a(h * w, T{}) {}
  inline T get(int i, int j) const { return a[w * i + j]; }
  inline void set(int i, int j, T v) { a[w * i + j] = v; }
  inline void add(int i, int j, T v) { a[w * i + j] += v; }
  inline void sub(int i, int j, T v) { a[w * i + j] -= v; }
  static Matrix id(int n) {
    Matrix mat(n);
    for (int i = 0; i < n; i++) mat.a[n * i + i] = T(1);
    return mat;
  }
  Matrix operator+=(const Matrix& r) {
    assert(h == r.h && w == r.w);
    for (int i = 0; i < h * w; i++) a[i] += r.a[i];
    return *this;
  }
  Matrix operator-=(const Matrix& r) {
    assert(h == r.h && w == r.w);
    for (int i = 0; i < h * w; i++) a[i] -= r.a[i];
    return *this;
  }
  Matrix operator+(const Matrix& r) { return Matrix(*this) += r; }
  Matrix operator-(const Matrix& r) { return Matrix(*this) -= r; }
  Matrix operator*(const Matrix& r) {
    assert(w == r.h);
    Matrix ret(h, r.w);
    for (int i = 0; i < h; i++)
      for (int j = 0; j < r.w; j++)
        for (int k = 0; k < w; k++)
          ret.add(i, j, get(i, k) * r.get(k, j));
    return ret;
  }
  Matrix& operator*=(const Matrix& r) { return *this = *this * r; }
  Matrix& operator*=(T r) {
    for (auto& v : a) v *= r;
    return *this;
  }
  Matrix operator*(T r) { return Matrix(*this) *= r; }
  Matrix pow(long long n) const {
    Matrix ret = id(h);
    Matrix mat(*this);
    while (n > 0) {
      if (n & 1) ret = ret * mat;
      mat = mat * mat;
      n >>= 1;
    }
    return ret;
  }

  T det() const {
    assert(h == w);
    Matrix mat(*this);
    T zero{}, det(1);
    for (int k = 0; k < h; k++) {
      {
        int i = k;
        while (i < h && mat.get(i, k) == zero) i++;
        if (i == h) return zero;
        if (i != k) {
          mat.swap_row(i, k);
          det = -det;
        }
      }
      for (int i = k + 1; i < h; i++)
        mat.mul_add_row(i, k, -mat.get(i, k) / mat.get(k, k));
      det *= mat.a[h * k + k];
    }
    return det;
  }
  optional<Matrix> inv() const {
    assert(h == w);
    Matrix mat(*this);
    Matrix imat = id(h);
    T zero{};
    for (int k = 0; k < h; k++) {
      {
        int i = k;
        while (i < h && mat.get(i, k) == zero) i++;
        if (i == h) return nullopt;
        if (i != k) {
          mat.swap_row(i, k);
          imat.swap_row(i, k);
        }
      }
      {
        T v = T(1) / mat.get(k, k);
        mat.mul_row(k, v);
        imat.mul_row(k, v);
      }
      for (int i = 0; i < h; i++) {
        if (i == k) continue;
        T v = -mat.get(i, k);
        mat.mul_add_row(i, k, v);
        imat.mul_add_row(i, k, v);
      }
    }
    return imat;
  }
  void swap_row(int i, int j) {
    for (int k = 0; k < w; k++) swap(a[w * i + k], a[w * j + k]);
  }
  void mul_row(int i, T v) {
    for (int k = 0; k < w; k++) a[w * i + k] *= v;
  }
  // row i += row j * v
  void mul_add_row(int i, int j, T v) {
    for (int k = 0; k < w; k++) a[w * i + k] += a[w * j + k] * v;
  }
  friend ostream& operator<<(ostream& os, const Matrix& mat) {
    for (int i = 0; i < mat.h; i++) {
      for (int j = 0; j < mat.w; j++) {
        os << mat.get(i, j);
        if (j + 1 < mat.w) os << " ";
      }
      if (i + 1 < mat.h) os << "\n";
    }
    return os;
  }
};
#line 5 "matrix/polynomial-matrix-prefix-product.hpp"

// a(k-1) a(k-2) ... a(0)
template <class mint>
Matrix<mint> PolynomialMatrixPrefixProduct(const Matrix<FormalPowerSeries<mint>>& a, long long k) {
  using mat = Matrix<mint>;
  assert(k >= 0 && a.h == a.w);
  int n = a.h, deg = 1;
  for (const auto& f : a.a) deg = max(deg, (int)f.size() - 1);
  while (deg & (deg - 1)) deg++;

  auto shift = [&](const vector<mat>& g, mint x) {
    int d = g.size();
    vector<mat> h(d, mat(n));
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < n; j++) {
        vector<mint> y(d);
        for (int l = 0; l < d; l++) y[l] = g[l].get(i, j);
        auto z = SamplingPointsShift(y, x, d);
        for (int l = 0; l < d; l++) h[l].set(i, j, z[l]);
      }
    }
    return h;
  };

  long long v = 1;
  while ((__int128)deg * v * v < k) v *= 2;
  vector<mat> g(deg + 1, mat(n));
  for (int i = 0; i <= deg; i++) {
    mint x = mint(v) * i;
    for (int j = 0; j < n; j++)
      for (int l = 0; l < n; l++) g[i].set(j, l, a.get(j, l).eval(x));
  }
  mint iv = mint(v).inv();
  for (long long w = 1; w < v; w *= 2) {
    auto g1 = shift(g, mint(w) * iv);
    auto g2 = shift(g, mint(w * deg * v + v) * iv);
    auto g3 = shift(g, mint(w * deg * v + v + w) * iv);
    for (int i = 0; i <= w * deg; i++) {
      g[i] = g1[i] * g[i];
      g2[i] = g3[i] * g2[i];
    }
    g.insert(g.end(), g2.begin(), g2.end() - 1);
  }

  mat ret = mat::id(n);
  long long i = 0;
  while (i + v <= k) {
    ret = g[i / v] * ret;
    i += v;
  }
  while (i < k) {
    mat m(n);
    for (int j = 0; j < n; j++)
      for (int l = 0; l < n; l++) m.set(j, l, a.get(j, l).eval(mint(i)));
    ret = m * ret;
    i++;
  }
  return ret;
}

/**
 * @brief 多項式行列の prefix product
 */
#line 4 "fps/p-recursive.hpp"

namespace p_recursive {

template <class T>
vector<T> NullVector(vector<vector<T>> a) {
  int h = a.size(), w = a[0].size(), rank = 0;
  vector<int> pivot;
  for (int j = 0; j < w && rank < h; j++) {
    int k = rank;
    while (k < h && a[k][j] == T{}) k++;
    if (k == h) continue;
    swap(a[rank], a[k]);
    T iv = T(1) / a[rank][j];
    for (int l = j; l < w; l++) a[rank][l] *= iv;
    for (int i = 0; i < h; i++) {
      if (i == rank || a[i][j] == T{}) continue;
      T c = a[i][j];
      for (int l = j; l < w; l++) a[i][l] -= c * a[rank][l];
    }
    pivot.push_back(j);
    rank++;
  }
  if (rank == w) return {};
  vector<bool> is_pivot(w, false);
  for (int j : pivot) is_pivot[j] = true;
  int free = 0;
  while (is_pivot[free]) free++;
  vector<T> x(w);
  x[free] = 1;
  for (int i = 0; i < rank; i++) x[pivot[i]] = -a[i][free];
  return x;
}

}  // namespace p_recursive

// sum[j=0...r] f[j](i) a[i+j] = 0 を満たす f を降順に返す
template <class mint>
vector<FormalPowerSeries<mint>> FindPRecursive(const vector<mint>& a, int d) {
  using fps = FormalPowerSeries<mint>;
  assert(d >= 0);
  int n = a.size();
  int r = (n + 2) / (d + 2) - 1;
  if (r <= 0) return {};
  int m = (r + 1) * (d + 1);
  vector<vector<mint>> mat(m - 1, vector<mint>(m));
  for (int i = 0; i < m - 1; i++) {
    for (int j = 0; j <= r; j++) {
      mint x = 1;
      for (int l = 0; l <= d; l++) {
        mat[i][(d + 1) * j + l] = x * a[i + j];
        x *= i + j;
      }
    }
  }
  auto c = p_recursive::NullVector(mat);
  if (c.empty()) return {};
  while ((int)c.size() > d + 1 &&
         all_of(c.end() - d - 1, c.end(), [](mint x) { return x == mint(0); }))
    c.erase(c.end() - d - 1, c.end());
  vector<fps> ret;
  for (int i = 0, j = 0; i < (int)c.size(); i += d + 1, j++) {
    fps f{1}, base{mint(j), mint(1)}, sum;
    for (int l = 0; l <= d; l++) {
      sum += f * c[i + l];
      f *= base;
    }
    sum.shrink();
    ret.push_back(sum);
  }
  reverse(ret.begin(), ret.end());
  return ret;
}

template <class mint>
mint PRecursiveTerm(const vector<mint>& a, long long k, int d) {
  if (k < (long long)a.size()) return a[k];
  if (all_of(a.begin(), a.end(), [](mint x) { return x == mint(0); })) return 0;
  auto f = FindPRecursive(a, d);
  assert(f.size() >= 2);
  int r = f.size() - 1;
  Matrix<FormalPowerSeries<mint>> m(r), denom(1);
  for (int i = 0; i < r; i++) m.set(0, i, -f[i + 1]);
  for (int i = 1; i < r; i++) m.set(i, i - 1, f[0]);
  denom.set(0, 0, f[0]);
  Matrix<mint> init(r, 1);
  for (int i = 0; i < r; i++) init.set(i, 0, a[r - 1 - i]);
  mint ret = (PolynomialMatrixPrefixProduct(m, k - r + 1) * init).get(0, 0);
  ret /= PolynomialMatrixPrefixProduct(denom, k - r + 1).get(0, 0);
  return ret;
}

template <class mint>
vector<mint> EnumeratePRecursive(const vector<mint>& a, long long n, int d) {
  assert(n >= 0);
  if (n <= (long long)a.size()) return vector<mint>(a.begin(), a.begin() + n);
  if (all_of(a.begin(), a.end(), [](mint x) { return x == mint(0); })) return vector<mint>(n);
  auto f = FindPRecursive(a, d);
  if (f.size() < 2) return {};
  int r = f.size() - 1;
  reverse(f.begin(), f.end());
  vector<mint> ret(a);
  ret.resize(n);
  for (long long k = a.size(); k < n; k++) {
    long long i = k - r;
    mint sum = 0;
    for (int j = 0; j < r; j++) sum += ret[i + j] * f[j].eval(mint(i));
    mint c = f[r].eval(mint(i));
    assert(c != mint(0));
    ret[k] = -sum / c;
  }
  return ret;
}

template <class mint>
mint PRecursiveTerm(const vector<mint>& a, long long k) {
  if (k < (long long)a.size()) return a[k];
  if (all_of(a.begin(), a.end(), [](mint x) { return x == mint(0); })) return 0;
  assert(a.size() >= 2);
  vector<mint> b(a.begin(), a.end() - 1);
  int n = a.size() - 1;
  for (int d = 0; (n + 2) / (d + 2) > 1; d++) {
    auto f = FindPRecursive(b, d);
    if (f.size() < 2) continue;
    auto c = EnumeratePRecursive(b, n + 1, d);
    if (!c.empty() && c.back() == a.back()) return PRecursiveTerm(a, k, d);
  }
  assert(false);
  return 0;
}

/**
 * @brief P-recursive
 * @docs docs/fps/p-recursive.md
 */

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