Eulerian number 形式的冪級数版
(include/emthrm/math/formal_power_series/eulerian_number_by_fps.hpp)

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Last update: 2023-05-13 18:14:57+09:00
Include: #include "emthrm/math/formal_power_series/eulerian_number_by_fps.hpp"

Eulerian number

\[A_n(x) = \sum_{m = 0}^n A(n, m) x^m\]

で定義される $A(n, m)$ である。ただし $A_n(x)$ は

\[\sum_{n = 0}^{\infty} A_n(x) \dfrac{t^n}{t!} = \dfrac{x - 1}{x - e^{(x - 1)t}}\]

で定義される Eulerian polynomials である。

\[A(n, m) = \begin{cases} 1 & (m = 0), \\ 0 & (n = m > 0), \\ (n - m) A(n - 1, m - 1) + (m + 1) A(n - 1, m) & (0 < m < n) \end{cases}\]

という漸化式をもつ。

一般項

\[A(n, m) = \sum_{k = 0}^m (-1)^k \binom{n + 1}{k} (m + 1 - k)^n\]

である。

時間計算量

	時間計算量
	$O(NM)$
形式的冪級数版	$O(N\log{N})$

仕様

名前	戻り値
`template <typename T>` `std::vector<std::vector<T>> eulerian_number(const int n, const int m);`	Eulerian number $A(i, j)$ ($0 \leq i \leq n,\ 0 \leq j \leq m$) の数表

形式的冪級数版

名前	戻り値
`template <unsigned int T>` `std::vector<MInt<T>> eulerian_number_init_by_fps(const int n);`	Eulerian number $A(n, j)$ ($0 \leq j \leq n$) の数表

参考文献

Leonhard Euler: Institutiones calculi differentialis (1755).
https://en.wikipedia.org/wiki/Eulerian_number
http://oeis.org/wiki/Eulerian_numbers,_triangle_of
~~https://min-25.hatenablog.com/entry/2015/04/07/160154~~

Depends on

Code

#ifndef EMTHRM_MATH_FORMAL_POWER_SERIES_EULERIAN_NUMBER_BY_FPS_HPP_
#define EMTHRM_MATH_FORMAL_POWER_SERIES_EULERIAN_NUMBER_BY_FPS_HPP_

#include <vector>

#include "emthrm/math/formal_power_series/formal_power_series.hpp"
#include "emthrm/math/modint.hpp"

namespace emthrm {

template <unsigned int T>
std::vector<MInt<T>> eulerian_number_init_by_fps(const int n) {
  using ModInt = MInt<T>;
  if (n == 0) [[unlikely]] return {1};
  ModInt::init(n + 1);
  const int m = (n + 1) / 2;
  FormalPowerSeries<ModInt> a(m - 1), b(m - 1);
  for (int i = 0; i < m; ++i) {
    a[i] = ModInt(i + 1).pow(n);
  }
  for (int i = 0; i < m; ++i) {
    b[i] = (i & 1 ? -ModInt::fact_inv(i) : ModInt::fact_inv(i))
           * ModInt::fact_inv(n + 1 - i);
  }
  a *= b;
  a.resize(n);
  for (int i = 0; i < m; ++i) {
    a[i] *= ModInt::fact(n + 1);
    a[n - 1 - i] = a[i];
  }
  return a.coef;
}

}  // namespace emthrm

#endif  // EMTHRM_MATH_FORMAL_POWER_SERIES_EULERIAN_NUMBER_BY_FPS_HPP_

#line 1 "include/emthrm/math/formal_power_series/eulerian_number_by_fps.hpp"



#include <vector>

#line 1 "include/emthrm/math/formal_power_series/formal_power_series.hpp"



#include <algorithm>
#include <cassert>
#include <functional>
#include <initializer_list>
#include <iterator>
#include <numeric>
#line 11 "include/emthrm/math/formal_power_series/formal_power_series.hpp"

namespace emthrm {

template <typename T>
struct FormalPowerSeries {
  std::vector<T> coef;

  explicit FormalPowerSeries(const int deg = 0) : coef(deg + 1, 0) {}
  explicit FormalPowerSeries(const std::vector<T>& coef) : coef(coef) {}
  FormalPowerSeries(const std::initializer_list<T> init)
      : coef(init.begin(), init.end()) {}
  template <typename InputIter>
  explicit FormalPowerSeries(const InputIter first, const InputIter last)
      : coef(first, last) {}

  inline const T& operator[](const int term) const { return coef[term]; }
  inline T& operator[](const int term) { return coef[term]; }

  using Mult = std::function<std::vector<T>(const std::vector<T>&,
                                            const std::vector<T>&)>;
  using Sqrt = std::function<bool(const T&, T*)>;
  static void set_mult(const Mult mult) { get_mult() = mult; }
  static void set_sqrt(const Sqrt sqrt) { get_sqrt() = sqrt; }

  void resize(const int deg) { coef.resize(deg + 1, 0); }
  void shrink() {
    while (coef.size() > 1 && coef.back() == 0) coef.pop_back();
  }
  int degree() const { return std::ssize(coef) - 1; }

  FormalPowerSeries& operator=(const std::vector<T>& coef_) {
    coef = coef_;
    return *this;
  }
  FormalPowerSeries& operator=(const FormalPowerSeries& x) = default;

  FormalPowerSeries& operator+=(const FormalPowerSeries& x) {
    const int deg_x = x.degree();
    if (deg_x > degree()) resize(deg_x);
    for (int i = 0; i <= deg_x; ++i) {
      coef[i] += x[i];
    }
    return *this;
  }
  FormalPowerSeries& operator-=(const FormalPowerSeries& x) {
    const int deg_x = x.degree();
    if (deg_x > degree()) resize(deg_x);
    for (int i = 0; i <= deg_x; ++i) {
      coef[i] -= x[i];
    }
    return *this;
  }
  FormalPowerSeries& operator*=(const T x) {
    for (T& e : coef) e *= x;
    return *this;
  }
  FormalPowerSeries& operator*=(const FormalPowerSeries& x) {
    return *this = get_mult()(coef, x.coef);
  }
  FormalPowerSeries& operator/=(const T x) {
    assert(x != 0);
    return *this *= static_cast<T>(1) / x;
  }
  FormalPowerSeries& operator/=(const FormalPowerSeries& x) {
    const int n = degree() - x.degree() + 1;
    if (n <= 0) return *this = FormalPowerSeries();
    const std::vector<T> tmp = get_mult()(
        std::vector<T>(coef.rbegin(), std::next(coef.rbegin(), n)),
        FormalPowerSeries(
            x.coef.rbegin(),
            std::next(x.coef.rbegin(), std::min(x.degree() + 1, n)))
        .inv(n - 1).coef);
    return *this = FormalPowerSeries(std::prev(tmp.rend(), n), tmp.rend());
  }
  FormalPowerSeries& operator%=(const FormalPowerSeries& x) {
    if (x.degree() == 0) return *this = FormalPowerSeries{0};
    *this -= *this / x * x;
    resize(x.degree() - 1);
    return *this;
  }
  FormalPowerSeries& operator<<=(const int n) {
    coef.insert(coef.begin(), n, 0);
    return *this;
  }
  FormalPowerSeries& operator>>=(const int n) {
    if (degree() < n) return *this = FormalPowerSeries();
    coef.erase(coef.begin(), coef.begin() + n);
    return *this;
  }

  bool operator==(FormalPowerSeries x) const {
    x.shrink();
    FormalPowerSeries y = *this;
    y.shrink();
    return x.coef == y.coef;
  }

  FormalPowerSeries operator+() const { return *this; }
  FormalPowerSeries operator-() const {
    FormalPowerSeries res = *this;
    for (T& e : res.coef) e = -e;
    return res;
  }

  FormalPowerSeries operator+(const FormalPowerSeries& x) const {
    return FormalPowerSeries(*this) += x;
  }
  FormalPowerSeries operator-(const FormalPowerSeries& x) const {
    return FormalPowerSeries(*this) -= x;
  }
  FormalPowerSeries operator*(const T x) const {
    return FormalPowerSeries(*this) *= x;
  }
  FormalPowerSeries operator*(const FormalPowerSeries& x) const {
    return FormalPowerSeries(*this) *= x;
  }
  FormalPowerSeries operator/(const T x) const {
    return FormalPowerSeries(*this) /= x;
  }
  FormalPowerSeries operator/(const FormalPowerSeries& x) const {
    return FormalPowerSeries(*this) /= x;
  }
  FormalPowerSeries operator%(const FormalPowerSeries& x) const {
    return FormalPowerSeries(*this) %= x;
  }
  FormalPowerSeries operator<<(const int n) const {
    return FormalPowerSeries(*this) <<= n;
  }
  FormalPowerSeries operator>>(const int n) const {
    return FormalPowerSeries(*this) >>= n;
  }

  T horner(const T x) const {
    return std::accumulate(
        coef.rbegin(), coef.rend(), static_cast<T>(0),
        [x](const T l, const T r) -> T { return l * x + r; });
  }

  FormalPowerSeries differential() const {
    const int deg = degree();
    assert(deg >= 0);
    FormalPowerSeries res(std::max(deg - 1, 0));
    for (int i = 1; i <= deg; ++i) {
      res[i - 1] = coef[i] * i;
    }
    return res;
  }

  FormalPowerSeries exp(const int deg) const {
    assert(coef[0] == 0);
    const int n = coef.size();
    const FormalPowerSeries one{1};
    FormalPowerSeries res = one;
    for (int i = 1; i <= deg; i <<= 1) {
      res *= FormalPowerSeries(coef.begin(),
                               std::next(coef.begin(), std::min(n, i << 1)))
             - res.log((i << 1) - 1) + one;
      res.coef.resize(i << 1);
    }
    res.resize(deg);
    return res;
  }
  FormalPowerSeries exp() const { return exp(degree()); }

  FormalPowerSeries inv(const int deg) const {
    assert(coef[0] != 0);
    const int n = coef.size();
    FormalPowerSeries res{static_cast<T>(1) / coef[0]};
    for (int i = 1; i <= deg; i <<= 1) {
      res = res + res - res * res * FormalPowerSeries(
          coef.begin(), std::next(coef.begin(), std::min(n, i << 1)));
      res.coef.resize(i << 1);
    }
    res.resize(deg);
    return res;
  }
  FormalPowerSeries inv() const { return inv(degree()); }

  FormalPowerSeries log(const int deg) const {
    assert(coef[0] == 1);
    FormalPowerSeries integrand = differential() * inv(deg - 1);
    integrand.resize(deg);
    for (int i = deg; i > 0; --i) {
      integrand[i] = integrand[i - 1] / i;
    }
    integrand[0] = 0;
    return integrand;
  }
  FormalPowerSeries log() const { return log(degree()); }

  FormalPowerSeries pow(long long exponent, const int deg) const {
    const int n = coef.size();
    if (exponent == 0) {
      FormalPowerSeries res(deg);
      if (deg != -1) [[unlikely]] res[0] = 1;
      return res;
    }
    assert(deg >= 0);
    for (int i = 0; i < n; ++i) {
      if (coef[i] == 0) continue;
      if (i > deg / exponent) break;
      const long long shift = exponent * i;
      T tmp = 1, base = coef[i];
      for (long long e = exponent; e > 0; e >>= 1) {
        if (e & 1) tmp *= base;
        base *= base;
      }
      const FormalPowerSeries res = ((*this >> i) / coef[i]).log(deg - shift);
      return ((res * exponent).exp(deg - shift) * tmp) << shift;
    }
    return FormalPowerSeries(deg);
  }
  FormalPowerSeries pow(const long long exponent) const {
    return pow(exponent, degree());
  }

  FormalPowerSeries mod_pow(long long exponent,
                            const FormalPowerSeries& md) const {
    const int deg = md.degree() - 1;
    if (deg < 0) [[unlikely]] return FormalPowerSeries(-1);
    const FormalPowerSeries inv_rev_md =
        FormalPowerSeries(md.coef.rbegin(), md.coef.rend()).inv();
    const auto mod_mult = [&md, &inv_rev_md, deg](
        FormalPowerSeries* multiplicand, const FormalPowerSeries& multiplier)
        -> void {
      *multiplicand *= multiplier;
      if (deg < multiplicand->degree()) {
        const int n = multiplicand->degree() - deg;
        const FormalPowerSeries quotient =
            FormalPowerSeries(multiplicand->coef.rbegin(),
                              std::next(multiplicand->coef.rbegin(), n))
            * FormalPowerSeries(
                  inv_rev_md.coef.begin(),
                  std::next(inv_rev_md.coef.begin(), std::min(deg + 2, n)));
        *multiplicand -=
            FormalPowerSeries(std::prev(quotient.coef.rend(), n),
                              quotient.coef.rend()) * md;
        multiplicand->resize(deg);
      }
      multiplicand->shrink();
    };
    FormalPowerSeries res{1}, base = *this;
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) mod_mult(&res, base);
      mod_mult(&base, base);
    }
    return res;
  }

  FormalPowerSeries sqrt(const int deg) const {
    const int n = coef.size();
    if (coef[0] == 0) {
      for (int i = 1; i < n; ++i) {
        if (coef[i] == 0) continue;
        if (i & 1) return FormalPowerSeries(-1);
        const int shift = i >> 1;
        if (deg < shift) break;
        FormalPowerSeries res = (*this >> i).sqrt(deg - shift);
        if (res.coef.empty()) return FormalPowerSeries(-1);
        res <<= shift;
        res.resize(deg);
        return res;
      }
      return FormalPowerSeries(deg);
    }
    T s;
    if (!get_sqrt()(coef.front(), &s)) return FormalPowerSeries(-1);
    FormalPowerSeries res{s};
    const T half = static_cast<T>(1) / 2;
    for (int i = 1; i <= deg; i <<= 1) {
      res = (FormalPowerSeries(coef.begin(),
                               std::next(coef.begin(), std::min(n, i << 1)))
             * res.inv((i << 1) - 1) + res) * half;
    }
    res.resize(deg);
    return res;
  }
  FormalPowerSeries sqrt() const { return sqrt(degree()); }

  FormalPowerSeries translate(const T c) const {
    const int n = coef.size();
    std::vector<T> fact(n, 1), inv_fact(n, 1);
    for (int i = 1; i < n; ++i) {
      fact[i] = fact[i - 1] * i;
    }
    inv_fact[n - 1] = static_cast<T>(1) / fact[n - 1];
    for (int i = n - 1; i > 0; --i) {
      inv_fact[i - 1] = inv_fact[i] * i;
    }
    std::vector<T> g(n), ex(n);
    for (int i = 0; i < n; ++i) {
      g[i] = coef[i] * fact[i];
    }
    std::reverse(g.begin(), g.end());
    T pow_c = 1;
    for (int i = 0; i < n; ++i) {
      ex[i] = pow_c * inv_fact[i];
      pow_c *= c;
    }
    const std::vector<T> conv = get_mult()(g, ex);
    FormalPowerSeries res(n - 1);
    for (int i = 0; i < n; ++i) {
      res[i] = conv[n - 1 - i] * inv_fact[i];
    }
    return res;
  }

 private:
  static Mult& get_mult() {
    static Mult mult = [](const std::vector<T>& a, const std::vector<T>& b)
        -> std::vector<T> {
      const int n = a.size(), m = b.size();
      std::vector<T> res(n + m - 1, 0);
      for (int i = 0; i < n; ++i) {
        for (int j = 0; j < m; ++j) {
          res[i + j] += a[i] * b[j];
        }
      }
      return res;
    };
    return mult;
  }
  static Sqrt& get_sqrt() {
    static Sqrt sqrt = [](const T&, T*) -> bool { return false; };
    return sqrt;
  }
};

}  // namespace emthrm


#line 1 "include/emthrm/math/modint.hpp"



#ifndef ARBITRARY_MODINT
#line 6 "include/emthrm/math/modint.hpp"
#endif
#include <compare>
#include <iostream>
// #include <numeric>
#include <utility>
#line 12 "include/emthrm/math/modint.hpp"

namespace emthrm {

#ifndef ARBITRARY_MODINT
template <unsigned int M>
struct MInt {
  unsigned int v;

  constexpr MInt() : v(0) {}
  constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {}
  static constexpr MInt raw(const int x) {
    MInt x_;
    x_.v = x;
    return x_;
  }

  static constexpr int get_mod() { return M; }
  static constexpr void set_mod(const int divisor) {
    assert(std::cmp_equal(divisor, M));
  }

  static void init(const int x) {
    inv<true>(x);
    fact(x);
    fact_inv(x);
  }

  template <bool MEMOIZES = false>
  static MInt inv(const int n) {
    // assert(0 <= n && n < M && std::gcd(n, M) == 1);
    static std::vector<MInt> inverse{0, 1};
    const int prev = inverse.size();
    if (n < prev) return inverse[n];
    if constexpr (MEMOIZES) {
      // "n!" and "M" must be disjoint.
      inverse.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        inverse[i] = -inverse[M % i] * raw(M / i);
      }
      return inverse[n];
    }
    int u = 1, v = 0;
    for (unsigned int a = n, b = M; b;) {
      const unsigned int q = a / b;
      std::swap(a -= q * b, b);
      std::swap(u -= q * v, v);
    }
    return u;
  }

  static MInt fact(const int n) {
    static std::vector<MInt> factorial{1};
    if (const int prev = factorial.size(); n >= prev) {
      factorial.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        factorial[i] = factorial[i - 1] * i;
      }
    }
    return factorial[n];
  }

  static MInt fact_inv(const int n) {
    static std::vector<MInt> f_inv{1};
    if (const int prev = f_inv.size(); n >= prev) {
      f_inv.resize(n + 1);
      f_inv[n] = inv(fact(n).v);
      for (int i = n; i > prev; --i) {
        f_inv[i - 1] = f_inv[i] * i;
      }
    }
    return f_inv[n];
  }

  static MInt nCk(const int n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
                                  fact_inv(n - k) * fact_inv(k));
  }
  static MInt nPk(const int n, const int k) {
    return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
  }
  static MInt nHk(const int n, const int k) {
    return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
  }

  static MInt large_nCk(long long n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    inv<true>(k);
    MInt res = 1;
    for (int i = 1; i <= k; ++i) {
      res *= inv(i) * n--;
    }
    return res;
  }

  constexpr MInt pow(long long exponent) const {
    MInt res = 1, tmp = *this;
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
    }
    return res;
  }

  constexpr MInt& operator+=(const MInt& x) {
    if ((v += x.v) >= M) v -= M;
    return *this;
  }
  constexpr MInt& operator-=(const MInt& x) {
    if ((v += M - x.v) >= M) v -= M;
    return *this;
  }
  constexpr MInt& operator*=(const MInt& x) {
    v = (unsigned long long){v} * x.v % M;
    return *this;
  }
  MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }

  constexpr auto operator<=>(const MInt& x) const = default;

  constexpr MInt& operator++() {
    if (++v == M) [[unlikely]] v = 0;
    return *this;
  }
  constexpr MInt operator++(int) {
    const MInt res = *this;
    ++*this;
    return res;
  }
  constexpr MInt& operator--() {
    v = (v == 0 ? M - 1 : v - 1);
    return *this;
  }
  constexpr MInt operator--(int) {
    const MInt res = *this;
    --*this;
    return res;
  }

  constexpr MInt operator+() const { return *this; }
  constexpr MInt operator-() const { return raw(v ? M - v : 0); }

  constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; }
  constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
  constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
  MInt operator/(const MInt& x) const { return MInt(*this) /= x; }

  friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
    return os << x.v;
  }
  friend std::istream& operator>>(std::istream& is, MInt& x) {
    long long v;
    is >> v;
    x = MInt(v);
    return is;
  }
};
#else  // ARBITRARY_MODINT
template <int ID>
struct MInt {
  unsigned int v;

  constexpr MInt() : v(0) {}
  MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {}
  static constexpr MInt raw(const int x) {
    MInt x_;
    x_.v = x;
    return x_;
  }

  static int get_mod() { return mod(); }
  static void set_mod(const unsigned int divisor) { mod() = divisor; }

  static void init(const int x) {
    inv<true>(x);
    fact(x);
    fact_inv(x);
  }

  template <bool MEMOIZES = false>
  static MInt inv(const int n) {
    // assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1);
    static std::vector<MInt> inverse{0, 1};
    const int prev = inverse.size();
    if (n < prev) return inverse[n];
    if constexpr (MEMOIZES) {
      // "n!" and "M" must be disjoint.
      inverse.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        inverse[i] = -inverse[mod() % i] * raw(mod() / i);
      }
      return inverse[n];
    }
    int u = 1, v = 0;
    for (unsigned int a = n, b = mod(); b;) {
      const unsigned int q = a / b;
      std::swap(a -= q * b, b);
      std::swap(u -= q * v, v);
    }
    return u;
  }

  static MInt fact(const int n) {
    static std::vector<MInt> factorial{1};
    if (const int prev = factorial.size(); n >= prev) {
      factorial.resize(n + 1);
      for (int i = prev; i <= n; ++i) {
        factorial[i] = factorial[i - 1] * i;
      }
    }
    return factorial[n];
  }

  static MInt fact_inv(const int n) {
    static std::vector<MInt> f_inv{1};
    if (const int prev = f_inv.size(); n >= prev) {
      f_inv.resize(n + 1);
      f_inv[n] = inv(fact(n).v);
      for (int i = n; i > prev; --i) {
        f_inv[i - 1] = f_inv[i] * i;
      }
    }
    return f_inv[n];
  }

  static MInt nCk(const int n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
                                  fact_inv(n - k) * fact_inv(k));
  }
  static MInt nPk(const int n, const int k) {
    return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
  }
  static MInt nHk(const int n, const int k) {
    return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
  }

  static MInt large_nCk(long long n, const int k) {
    if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
    inv<true>(k);
    MInt res = 1;
    for (int i = 1; i <= k; ++i) {
      res *= inv(i) * n--;
    }
    return res;
  }

  MInt pow(long long exponent) const {
    MInt res = 1, tmp = *this;
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
    }
    return res;
  }

  MInt& operator+=(const MInt& x) {
    if ((v += x.v) >= mod()) v -= mod();
    return *this;
  }
  MInt& operator-=(const MInt& x) {
    if ((v += mod() - x.v) >= mod()) v -= mod();
    return *this;
  }
  MInt& operator*=(const MInt& x) {
    v = (unsigned long long){v} * x.v % mod();
    return *this;
    }
  MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }

  auto operator<=>(const MInt& x) const = default;

  MInt& operator++() {
    if (++v == mod()) [[unlikely]] v = 0;
    return *this;
  }
  MInt operator++(int) {
    const MInt res = *this;
    ++*this;
    return res;
  }
  MInt& operator--() {
    v = (v == 0 ? mod() - 1 : v - 1);
    return *this;
  }
  MInt operator--(int) {
    const MInt res = *this;
    --*this;
    return res;
  }

  MInt operator+() const { return *this; }
  MInt operator-() const { return raw(v ? mod() - v : 0); }

  MInt operator+(const MInt& x) const { return MInt(*this) += x; }
  MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
  MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
  MInt operator/(const MInt& x) const { return MInt(*this) /= x; }

  friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
    return os << x.v;
  }
  friend std::istream& operator>>(std::istream& is, MInt& x) {
    long long v;
    is >> v;
    x = MInt(v);
    return is;
  }

 private:
  static unsigned int& mod() {
    static unsigned int divisor = 0;
    return divisor;
  }
};
#endif  // ARBITRARY_MODINT

}  // namespace emthrm


#line 8 "include/emthrm/math/formal_power_series/eulerian_number_by_fps.hpp"

namespace emthrm {

template <unsigned int T>
std::vector<MInt<T>> eulerian_number_init_by_fps(const int n) {
  using ModInt = MInt<T>;
  if (n == 0) [[unlikely]] return {1};
  ModInt::init(n + 1);
  const int m = (n + 1) / 2;
  FormalPowerSeries<ModInt> a(m - 1), b(m - 1);
  for (int i = 0; i < m; ++i) {
    a[i] = ModInt(i + 1).pow(n);
  }
  for (int i = 0; i < m; ++i) {
    b[i] = (i & 1 ? -ModInt::fact_inv(i) : ModInt::fact_inv(i))
           * ModInt::fact_inv(n + 1 - i);
  }
  a *= b;
  a.resize(n);
  for (int i = 0; i < m; ++i) {
    a[i] *= ModInt::fact(n + 1);
    a[n - 1 - i] = a[i];
  }
  return a.coef;
}

}  // namespace emthrm

Eulerian number 形式的冪級数版 (include/emthrm/math/formal_power_series/eulerian_number_by_fps.hpp)