fps/fps-arbitrary.hpp

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• Last update: 2025-12-29 01:13:32+09:00
• Include: #include "fps/fps-arbitrary.hpp"

Depends on

• 任意 mod 畳み込み (convolution/intmod.hpp)
• NTT (数論変換) (fft/ntt.hpp)
• fps/formal-power-series.hpp
• math/util.hpp
• modint/modint.hpp

Code

#pragma once

#include "convolution/intmod.hpp"
#include "fps/formal-power-series.hpp"

template <class mint>
void FormalPowerSeries<mint>::set_ntt() { ntt_ptr = nullptr; }
template <class mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FormalPowerSeries<mint>& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  auto ret = ConvolutionIntMod::multiply(*this, r);
  return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::middle_product(const FormalPowerSeries<mint>& r) const {
  auto ret = ConvolutionIntMod::middle_product(*this, r);
  return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
void FormalPowerSeries<mint>::ntt() { exit(1); }
template <class mint>
void FormalPowerSeries<mint>::intt() { exit(1); }
template <class mint>
void FormalPowerSeries<mint>::ntt_doubling() { exit(1); }
template <typename mint>
int FormalPowerSeries<mint>::ntt_root() { exit(1); }
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  assert((*this)[0] != mint(0));
  if (deg == -1) deg = (*this).size();
  FPS ret{mint(1) / (*this)[0]};
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
  return ret.pre(deg);
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  assert((*this)[0] == mint(0));
  if (deg == -1) deg = (*this).size();
  FPS ret{mint(1)};
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret * ((*this).pre(i << 1) - ret.log(i << 1) + 1)).pre(i << 1);
  return ret.pre(deg);
}

#line 2 "fps/fps-arbitrary.hpp"

#line 2 "convolution/intmod.hpp"

#line 2 "math/util.hpp"

namespace Math {
template <class T>
T safe_mod(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  a %= b;
  return a >= 0 ? a : a + b;
}
template <class T>
T floor(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a >= 0 ? a / b : (a + 1) / b - 1;
}
template <class T>
T ceil(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a > 0 ? (a - 1) / b + 1 : a / b;
}
long long isqrt(long long n) {
  if (n <= 0) return 0;
  long long x = sqrt(n);
  while ((x + 1) * (x + 1) <= n) x++;
  while (x * x > n) x--;
  return x;
}
// return g=gcd(a,b)
// a*x+b*y=g
// - b!=0 -> 0<=x<|b|/g
// - b=0  -> ax=g
template <class T>
T ext_gcd(T a, T b, T& x, T& y) {
  T a0 = a, b0 = b;
  bool sgn_a = a < 0, sgn_b = b < 0;
  if (sgn_a) a = -a;
  if (sgn_b) b = -b;
  if (b == 0) {
    x = sgn_a ? -1 : 1;
    y = 0;
    return a;
  }
  T x00 = 1, x01 = 0, x10 = 0, x11 = 1;
  while (b != 0) {
    T q = a / b, r = a - b * q;
    x00 -= q * x01;
    x10 -= q * x11;
    swap(x00, x01);
    swap(x10, x11);
    a = b, b = r;
  }
  x = x00, y = x10;
  if (sgn_a) x = -x;
  if (sgn_b) y = -y;
  if (b0 != 0) {
    a0 /= a, b0 /= a;
    if (b0 < 0) a0 = -a0, b0 = -b0;
    T q = x >= 0 ? x / b0 : (x + 1) / b0 - 1;
    x -= b0 * q;
    y += a0 * q;
  }
  return a;
}
constexpr long long inv_mod(long long x, long long m) {
  x %= m;
  if (x < 0) x += m;
  long long a = m, b = x;
  long long y0 = 0, y1 = 1;
  while (b > 0) {
    long long q = a / b;
    swap(a -= q * b, b);
    swap(y0 -= q * y1, y1);
  }
  if (y0 < 0) y0 += m / a;
  return y0;
}
long long pow_mod(long long x, long long n, long long m) {
  x = (x % m + m) % m;
  long long y = 1;
  while (n) {
    if (n & 1) y = y * x % m;
    x = x * x % m;
    n >>= 1;
  }
  return y;
}
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
  if (m == 1) return 0;
  unsigned int _m = (unsigned int)(m);
  unsigned long long r = 1;
  unsigned long long y = x % m;
  if (y >= m) y += m;
  while (n) {
    if (n & 1) r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}
constexpr bool is_prime_constexpr(int n) {
  if (n <= 1) return false;
  if (n == 2 || n == 7 || n == 61) return true;
  if (n % 2 == 0) return false;
  long long d = n - 1;
  while (d % 2 == 0) d /= 2;
  constexpr long long bases[3] = {2, 7, 61};
  for (long long a : bases) {
    long long t = d;
    long long y = pow_mod_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1) {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0) {
      return false;
    }
  }
  return true;
}
template <int n>
constexpr bool is_prime = is_prime_constexpr(n);
};  // namespace Math
#line 3 "modint/modint.hpp"

template <unsigned int m = 998244353>
struct ModInt {
  using mint = ModInt;
  static constexpr unsigned int get_mod() { return m; }
  static mint raw(int v) {
    mint x;
    x._v = v;
    return x;
  }
  ModInt() : _v(0) {}
  ModInt(int64_t v) {
    long long x = (long long)(v % (long long)(umod()));
    if (x < 0) x += umod();
    _v = (unsigned int)(x);
  }
  unsigned int val() const { return _v; }
  mint& operator++() {
    _v++;
    if (_v == umod()) _v = 0;
    return *this;
  }
  mint& operator--() {
    if (_v == 0) _v = umod();
    _v--;
    return *this;
  }
  mint operator++(int) {
    mint result = *this;
    ++*this;
    return result;
  }
  mint operator--(int) {
    mint result = *this;
    --*this;
    return result;
  }
  mint& operator+=(const mint& rhs) {
    _v += rhs._v;
    if (_v >= umod()) _v -= umod();
    return *this;
  }
  mint& operator-=(const mint& rhs) {
    _v -= rhs._v;
    if (_v >= umod()) _v += umod();
    return *this;
  }
  mint& operator*=(const mint& rhs) {
    unsigned long long z = _v;
    z *= rhs._v;
    _v = (unsigned int)(z % umod());
    return *this;
  }
  mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
  mint operator+() const { return *this; }
  mint operator-() const { return mint() - *this; }
  mint pow(long long n) const {
    assert(0 <= n);
    mint x = *this, r = 1;
    while (n) {
      if (n & 1) r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  mint inv() const {
    if (is_prime) {
      assert(_v);
      return pow(umod() - 2);
    } else {
      auto inv = Math::inv_mod(_v, umod());
      return raw(inv);
    }
  }
  friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; }
  friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; }
  friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; }
  friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; }
  friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
  friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
  friend istream& operator>>(istream& is, mint& x) {
    int64_t v;
    is >> v;
    x = mint(v);
    return is;
  }
  friend ostream& operator<<(ostream& os, const mint& x) { return os << x.val(); }

 private:
  unsigned int _v;
  static constexpr unsigned int umod() { return m; }
  static constexpr bool is_prime = Math::is_prime<m>;
};
#line 2 "fft/ntt.hpp"

template <class mint>
struct NTT {
  static constexpr unsigned int mod = mint::get_mod();
  static constexpr unsigned long long pow_constexpr(unsigned long long x, unsigned long long n, unsigned long long m) {
    unsigned long long y = 1;
    while (n) {
      if (n & 1) y = y * x % m;
      x = x * x % m;
      n >>= 1;
    }
    return y;
  }
  static constexpr unsigned int get_g() {
    unsigned long long x = 2;
    while (pow_constexpr(x, (mod - 1) >> 1, mod) == 1) x += 1;
    return x;
  }
  static constexpr unsigned int g = get_g();
  static constexpr int rank2 = __builtin_ctzll(mod - 1);
  array<mint, rank2 + 1> root;
  array<mint, rank2 + 1> iroot;
  array<mint, max(0, rank2 - 2 + 1)> rate2;
  array<mint, max(0, rank2 - 2 + 1)> irate2;
  array<mint, max(0, rank2 - 3 + 1)> rate3;
  array<mint, max(0, rank2 - 3 + 1)> irate3;

  NTT() {
    root[rank2] = mint(g).pow((mod - 1) >> rank2);
    iroot[rank2] = root[rank2].inv();
    for (int i = rank2 - 1; i >= 0; i--) {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 2; i++) {
        rate2[i] = root[i + 2] * prod;
        irate2[i] = iroot[i + 2] * iprod;
        prod *= iroot[i + 2];
        iprod *= root[i + 2];
      }
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 3; i++) {
        rate3[i] = root[i + 3] * prod;
        irate3[i] = iroot[i + 3] * iprod;
        prod *= iroot[i + 3];
        iprod *= root[i + 3];
      }
    }
  }
  void ntt(vector<mint>& a) {
    int n = int(a.size());
    int h = __builtin_ctzll((unsigned int)n);
    a.resize(1 << h);
    int len = 0;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len < h) {
      if (h - len == 1) {
        int p = 1 << (h - len - 1);
        mint rot = 1;
        for (int s = 0; s < (1 << len); s++) {
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            auto l = a[i + offset];
            auto r = a[i + offset + p] * rot;
            a[i + offset] = l + r;
            a[i + offset + p] = l - r;
          }
          if (s + 1 != (1 << len)) rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
        }
        len++;
      } else {
        // 4-base
        int p = 1 << (h - len - 2);
        mint rot = 1, imag = root[2];
        for (int s = 0; s < (1 << len); s++) {
          mint rot2 = rot * rot;
          mint rot3 = rot2 * rot;
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            auto mod2 = 1ULL * mint::get_mod() * mint::get_mod();
            auto a0 = 1ULL * a[i + offset].val();
            auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
            auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
            auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
            auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
            auto na2 = mod2 - a2;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
            a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
            a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
          }
          if (s + 1 != (1 << len)) rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
        }
        len += 2;
      }
    }
  }
  void intt(vector<mint>& a) {
    int n = int(a.size());
    int h = __builtin_ctzll((unsigned int)n);
    a.resize(1 << h);

    int len = h;  // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len) {
      if (len == 1) {
        int p = 1 << (h - len);
        mint irot = 1;
        for (int s = 0; s < (1 << (len - 1)); s++) {
          int offset = s << (h - len + 1);
          for (int i = 0; i < p; i++) {
            auto l = a[i + offset];
            auto r = a[i + offset + p];
            a[i + offset] = l + r;
            a[i + offset + p] = (unsigned long long)(mint::get_mod() + l.val() - r.val()) * irot.val();
          }
          if (s + 1 != (1 << (len - 1))) irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
        }
        len--;
      } else {
        // 4-base
        int p = 1 << (h - len);
        mint irot = 1, iimag = iroot[2];
        for (int s = 0; s < (1 << (len - 2)); s++) {
          mint irot2 = irot * irot;
          mint irot3 = irot2 * irot;
          int offset = s << (h - len + 2);
          for (int i = 0; i < p; i++) {
            auto a0 = 1ULL * a[i + offset + 0 * p].val();
            auto a1 = 1ULL * a[i + offset + 1 * p].val();
            auto a2 = 1ULL * a[i + offset + 2 * p].val();
            auto a3 = 1ULL * a[i + offset + 3 * p].val();
            auto a2na3iimag = 1ULL * mint((mint::get_mod() + a2 - a3) * iimag.val()).val();
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + 1 * p] = (a0 + (mint::get_mod() - a1) + a2na3iimag) * irot.val();
            a[i + offset + 2 * p] = (a0 + a1 + (mint::get_mod() - a2) + (mint::get_mod() - a3)) * irot2.val();
            a[i + offset + 3 * p] = (a0 + (mint::get_mod() - a1) + (mint::get_mod() - a2na3iimag)) * irot3.val();
          }
          if (s + 1 != (1 << (len - 2))) irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
        }
        len -= 2;
      }
    }
    mint e = mint(n).inv();
    for (auto& x : a) x *= e;
  }
  vector<mint> multiply(const vector<mint>& a, const vector<mint>& b) {
    if (a.empty() || b.empty()) return vector<mint>();
    int n = a.size(), m = b.size();
    int sz = n + m - 1;
    if (n <= 30 || m <= 30) {
      if (n > 30) return multiply(b, a);
      vector<mint> res(sz);
      for (int i = 0; i < n; i++)
        for (int j = 0; j < m; j++) res[i + j] += a[i] * b[j];
      return res;
    }
    int sz1 = 1;
    while (sz1 < sz) sz1 <<= 1;
    vector<mint> res(sz1);
    for (int i = 0; i < n; i++) res[i] = a[i];
    ntt(res);
    if (a == b)
      for (int i = 0; i < sz1; i++) res[i] *= res[i];
    else {
      vector<mint> c(sz1);
      for (int i = 0; i < m; i++) c[i] = b[i];
      ntt(c);
      for (int i = 0; i < sz1; i++) res[i] *= c[i];
    }
    intt(res);
    res.resize(sz);
    return res;
  }
  // c[i]=sum[j]a[j]b[i+j]
  vector<mint> middle_product(const vector<mint>& a, const vector<mint>& b) {
    if (b.empty() || a.size() > b.size()) return {};
    int n = a.size(), m = b.size();
    int sz = m - n + 1;
    if (n <= 30 || sz <= 30) {
      vector<mint> res(sz);
      for (int i = 0; i < sz; i++)
        for (int j = 0; j < n; j++) res[i] += a[j] * b[i + j];
      return res;
    }
    int sz1 = 1;
    while (sz1 < m) sz1 <<= 1;
    vector<mint> res(sz1), b2(sz1);
    reverse_copy(a.begin(), a.end(), res.begin());
    copy(b.begin(), b.end(), b2.begin());
    ntt(res);
    ntt(b2);
    for (int i = 0; i < res.size(); i++) res[i] *= b2[i];
    intt(res);
    res.resize(m);
    res.erase(res.begin(), res.begin() + n - 1);
    return res;
  }
  void ntt_doubling(vector<mint>& a) {
    int n = (int)a.size();
    auto b = a;
    intt(b);
    mint r = 1, zeta = mint(g).pow((mint::get_mod() - 1) / (n << 1));
    for (int i = 0; i < n; i++) b[i] *= r, r *= zeta;
    ntt(b);
    copy(b.begin(), b.end(), back_inserter(a));
  }
};
/**
 * @brief NTT (数論変換)
 * @docs docs/fft/ntt.md
 */
#line 5 "convolution/intmod.hpp"

namespace ConvolutionIntMod {
using ll = long long;
static constexpr ll Mod1 = 754974721;
static constexpr ll Mod2 = 167772161;
static constexpr ll Mod3 = 469762049;
static constexpr ll M1invM2 = 95869806;
static constexpr ll M12invM3 = 187290749;
using M1 = ModInt<Mod1>;
using M2 = ModInt<Mod2>;
using M3 = ModInt<Mod3>;
NTT<M1> ntt1;
NTT<M2> ntt2;
NTT<M3> ntt3;

template <class mint>
vector<mint> multiply(const vector<mint>& a, const vector<mint>& b) {
  if (a.empty() || b.empty()) return {};
  int mod = mint::get_mod();
  ll M12mod = Mod1 * Mod2 % mod;
  vector<unsigned int> a0(a.size()), b0(b.size());
  for (int i = 0; i < a.size(); i++) a0[i] = a[i].val();
  for (int i = 0; i < b.size(); i++) b0[i] = b[i].val();

  vector<M1> a1(a0.begin(), a0.end()), b1(b0.begin(), b0.end()), c1 = ntt1.multiply(a1, b1);
  vector<M2> a2(a0.begin(), a0.end()), b2(b0.begin(), b0.end()), c2 = ntt2.multiply(a2, b2);
  vector<M3> a3(a0.begin(), a0.end()), b3(b0.begin(), b0.end()), c3 = ntt3.multiply(a3, b3);
  vector<mint> c(a.size() + b.size() - 1, 0);
  for (int i = 0; i < c.size(); i++) {
    ll v1 = ((ll)c2[i].val() - (ll)c1[i].val()) * M1invM2 % Mod2;
    if (v1 < 0) v1 += Mod2;
    ll v2 = ((ll)c3[i].val() - ((ll)c1[i].val() + Mod1 * v1) % Mod3) * M12invM3 % Mod3;
    if (v2 < 0) v2 += Mod3;
    ll v3 = ((ll)c1[i].val() + Mod1 * v1 + M12mod * v2) % mod;
    if (v3 < 0) v3 += mod;
    c[i] = v3;
  }
  return c;
}
template <class mint>
vector<mint> middle_product(const vector<mint>& a, const vector<mint>& b) {
  if (b.empty() || a.size() > b.size()) return {};
  int mod = mint::get_mod();
  ll M12mod = Mod1 * Mod2 % mod;
  vector<unsigned int> a0(a.size()), b0(b.size());
  for (int i = 0; i < a.size(); i++) a0[i] = a[i].val();
  for (int i = 0; i < b.size(); i++) b0[i] = b[i].val();

  vector<M1> a1(a0.begin(), a0.end()), b1(b0.begin(), b0.end()), c1 = ntt1.middle_product(a1, b1);
  vector<M2> a2(a0.begin(), a0.end()), b2(b0.begin(), b0.end()), c2 = ntt2.middle_product(a2, b2);
  vector<M3> a3(a0.begin(), a0.end()), b3(b0.begin(), b0.end()), c3 = ntt3.middle_product(a3, b3);
  vector<mint> c(c1.size(), 0);
  for (int i = 0; i < c.size(); i++) {
    ll v1 = ((ll)c2[i].val() - (ll)c1[i].val()) * M1invM2 % Mod2;
    if (v1 < 0) v1 += Mod2;
    ll v2 = ((ll)c3[i].val() - ((ll)c1[i].val() + Mod1 * v1) % Mod3) * M12invM3 % Mod3;
    if (v2 < 0) v2 += Mod3;
    ll v3 = ((ll)c1[i].val() + Mod1 * v1 + M12mod * v2) % mod;
    if (v3 < 0) v3 += mod;
    c[i] = v3;
  }
  return c;
}
};  // namespace ConvolutionIntMod

/**
 * @brief 任意 mod 畳み込み
 * @docs docs/convolution/intmod.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 5 "fps/fps-arbitrary.hpp"

template <class mint>
void FormalPowerSeries<mint>::set_ntt() { ntt_ptr = nullptr; }
template <class mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FormalPowerSeries<mint>& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  auto ret = ConvolutionIntMod::multiply(*this, r);
  return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::middle_product(const FormalPowerSeries<mint>& r) const {
  auto ret = ConvolutionIntMod::middle_product(*this, r);
  return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
void FormalPowerSeries<mint>::ntt() { exit(1); }
template <class mint>
void FormalPowerSeries<mint>::intt() { exit(1); }
template <class mint>
void FormalPowerSeries<mint>::ntt_doubling() { exit(1); }
template <typename mint>
int FormalPowerSeries<mint>::ntt_root() { exit(1); }
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  assert((*this)[0] != mint(0));
  if (deg == -1) deg = (*this).size();
  FPS ret{mint(1) / (*this)[0]};
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
  return ret.pre(deg);
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  assert((*this)[0] == mint(0));
  if (deg == -1) deg = (*this).size();
  FPS ret{mint(1)};
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret * ((*this).pre(i << 1) - ret.log(i << 1) + 1)).pre(i << 1);
  return ret.pre(deg);
}

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