Formal Power Series Friendly NTT(NTTmod用形式的冪級数) (math/fps/formal-power-series-friendly-ntt.hpp)

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• Last update: 2024-04-27 01:27:28+09:00
• Include: #include "math/fps/formal-power-series-friendly-ntt.hpp"
• Link: https://judge.yosupo.jp/problem/convolution_mod
• Link: https://judge.yosupo.jp/problem/division_of_polynomials
• Link: https://judge.yosupo.jp/problem/exp_of_formal_power_series
• Link: https://judge.yosupo.jp/problem/inv_of_formal_power_series
• Link: https://judge.yosupo.jp/problem/log_of_formal_power_series
• Link: https://judge.yosupo.jp/problem/polynomial_taylor_shift
• Link: https://judge.yosupo.jp/problem/pow_of_formal_power_series
• Link: https://judge.yosupo.jp/problem/sqrt_of_formal_power_series

概要

NTT を利用可能な mod のもとで、形式的べき級数の各演算を効率的に行う。

TODO 解説

使い方

通常はテンプレートの型として Modint 構造体を渡す。また static な関数として ntt と intt が実装された構造体が必要である。この構造体は NumberTheoreticTransformFriendlyModInt< T > に対応しているが、他のライブラリを使いたい場合は必要に応じてこの部分を書き換えること。

計算量が書かれていない演算は $O(n \log n)$ で動作する。引数で deg を渡すことで, deg 項まで計算して返すものもある(デフォルトでは基本的に $\mathrm{deg}(f)$ 項で打ち切る)。

• +: $f(x) + g(x)$ を返す。$O(n)$
• -: $f(x) - g(x)$ を返す。$O(n)$
• *: $f(x) \times g(x)$ を返す。
• /: $f(x) = q(x) \times g(x) + r(x)$ かつ $\mathrm{deg}(r) \lt \mathrm{deg}(g)$ を満たす $q(x)$ を返す。
• %: $f(x) = q(x) \times g(x) + r(x)$ かつ $\mathrm{deg}(r) \lt \mathrm{deg}(g)$ を満たす $r(x)$ を返す。
• div_mod(): 剰余を返す。/ と % をそれぞれ呼び出すよりも効率的。
• operator(x): $f(x)$ の値を評価して返す。$O(n)$
• diff(): $f(x)’$ を返す。$O(n)$
• integral(): $\int f(x) dx$ を返す。$O(n)$
• inv(): $\frac {1} {f(x)}$ を返す。$f(0) \neq 0$ を要求する。
• log(): $\log f(x)$ を返す。$f(0) = 1$ を要求する。
• sqrt(get_sqrt): $\sqrt {f(x)}$、つまり $f(x) = g(x)^2$ を満たす $g(x)$ を返す。存在しない場合空配列を返す。get_sqrt はあるmodint $y$ が与えられたときに $y = x^2$ を満たす modint $x$ を返すラムダ式で, これを渡さない場合は非 $0$ な最初の項が $1$ であることを要求する。存在しない場合は適当な値を返すように実装すると、空配列が返される。
• exp(): $e^{f} (x)$ を返す。$f(0) = 0$ を要求する。
• pow(k): $f^{k} (x)$ を返す。
• mod_pow(k, g): $f^{k} (x) \pmod {g(x)}$ を返す。$O(n \log k \log \mathrm{deg}(f))$
• taylor_shift(c): $g(x) = f(x + c)$ を満たす $g(x)$ を返す。

Depends on

• Number Theoretic Transform Friendly ModInt (math/fft/number-theoretic-transform-friendly-mod-int.hpp)

Verified with

• test/verify/yosupo-bernoulli-number.test.cpp
• test/verify/yosupo-division-of-polynomials.test.cpp
• test/verify/yosupo-exp-of-formal-power-series.test.cpp
• test/verify/yosupo-find-linear-recurrence.test.cpp
• test/verify/yosupo-inv-of-formal-power-series.test.cpp
• test/verify/yosupo-kth-term-of-linearly-recurrent-sequence.test.cpp
• test/verify/yosupo-log-of-formal-power-series.test.cpp
• test/verify/yosupo-multipoint-evaluation.test.cpp
• test/verify/yosupo-partition-function.test.cpp
• test/verify/yosupo-polynomial-interpolation.test.cpp
• test/verify/yosupo-polynomial-taylor-shift.test.cpp
• test/verify/yosupo-pow-of-formal-power-series.test.cpp
• test/verify/yosupo-sharp-p-subset-sum.test.cpp
• test/verify/yosupo-sqrt-of-formal-power-series.test.cpp
• test/verify/yosupo-stirling-number-of-the-first-kind.test.cpp
• test/verify/yosupo-stirling-number-of-the-second-kind.test.cpp

Code

#pragma once

#include "../fft/number-theoretic-transform-friendly-mod-int.hpp"

template <typename T>
struct FormalPowerSeriesFriendlyNTT : vector<T> {
  using vector<T>::vector;
  using P = FormalPowerSeriesFriendlyNTT;
  using NTT = NumberTheoreticTransformFriendlyModInt<T>;

  P pre(int deg) const {
    return P(begin(*this), begin(*this) + min((int)this->size(), deg));
  }

  P rev(int deg = -1) const {
    P ret(*this);
    if (deg != -1) ret.resize(deg, T(0));
    reverse(begin(ret), end(ret));
    return ret;
  }

  void shrink() {
    while (this->size() && this->back() == T(0)) this->pop_back();
  }

  P operator+(const P &r) const { return P(*this) += r; }

  P operator+(const T &v) const { return P(*this) += v; }

  P operator-(const P &r) const { return P(*this) -= r; }

  P operator-(const T &v) const { return P(*this) -= v; }

  P operator*(const P &r) const { return P(*this) *= r; }

  P operator*(const T &v) const { return P(*this) *= v; }

  P operator/(const P &r) const { return P(*this) /= r; }

  P operator%(const P &r) const { return P(*this) %= r; }

  P &operator+=(const P &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;
  }

  P &operator-=(const P &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;
  }

  // https://judge.yosupo.jp/problem/convolution_mod
  P &operator*=(const P &r) {
    if (this->empty() || r.empty()) {
      this->clear();
      return *this;
    }
    auto ret = NTT::multiply(*this, r);
    return *this = {begin(ret), end(ret)};
  }

  P &operator/=(const P &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
  }

  P &operator%=(const P &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }

  // https://judge.yosupo.jp/problem/division_of_polynomials
  pair<P, P> div_mod(const P &r) {
    P q = *this / r;
    P x = *this - q * r;
    x.shrink();
    return make_pair(q, x);
  }

  P operator-() const {
    P ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  P &operator+=(const T &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

  P &operator-=(const T &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }

  P &operator*=(const T &v) {
    for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= v;
    return *this;
  }

  P dot(P r) const {
    P ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }

  P operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    P ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }

  P operator<<(int sz) const {
    P ret(*this);
    ret.insert(ret.begin(), sz, T(0));
    return ret;
  }

  T operator()(T x) const {
    T r = 0, w = 1;
    for (auto &v : *this) {
      r += w * v;
      w *= x;
    }
    return r;
  }

  P diff() const {
    const int n = (int)this->size();
    P ret(max(0, n - 1));
    for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
    return ret;
  }

  P integral() const {
    const int n = (int)this->size();
    P ret(n + 1);
    ret[0] = T(0);
    for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
    return ret;
  }

  // https://judge.yosupo.jp/problem/inv_of_formal_power_series
  // F(0) must not be 0
  P inv(int deg = -1) const {
    assert(((*this)[0]) != T(0));
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    P res(deg);
    res[0] = {T(1) / (*this)[0]};
    for (int d = 1; d < deg; d <<= 1) {
      P f(2 * d), g(2 * d);
      for (int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
      for (int j = 0; j < d; j++) g[j] = res[j];
      NTT::ntt(f);
      NTT::ntt(g);
      f = f.dot(g);
      NTT::intt(f);
      for (int j = 0; j < d; j++) f[j] = 0;
      NTT::ntt(f);
      for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
      NTT::intt(f);
      for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
    }
    return res;
  }

  // https://judge.yosupo.jp/problem/log_of_formal_power_series
  // F(0) must be 1
  P log(int deg = -1) const {
    assert((*this)[0] == T(1));
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
  P sqrt(
      int deg = -1,
      const function<T(T)> &get_sqrt = [](T) { return T(1); }) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if ((*this)[0] == T(0)) {
      for (int i = 1; i < n; i++) {
        if ((*this)[i] != T(0)) {
          if (i & 1) return {};
          if (deg - i / 2 <= 0) break;
          auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
          if (ret.empty()) return {};
          ret = ret << (i / 2);
          if ((int)ret.size() < deg) ret.resize(deg, T(0));
          return ret;
        }
      }
      return P(deg, 0);
    }
    auto sqr = T(get_sqrt((*this)[0]));
    if (sqr * sqr != (*this)[0]) return {};
    P ret{sqr};
    T inv2 = T(1) / T(2);
    for (int i = 1; i < deg; i <<= 1) {
      ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
    }
    return ret.pre(deg);
  }

  P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
    return sqrt(deg, get_sqrt);
  }

  // https://judge.yosupo.jp/problem/exp_of_formal_power_series
  // F(0) must be 0
  P exp(int deg = -1) const {
    if (deg == -1) deg = this->size();
    assert((*this)[0] == T(0));

    P inv;
    inv.reserve(deg + 1);
    inv.push_back(T(0));
    inv.push_back(T(1));

    auto inplace_integral = [&](P &F) -> void {
      const int n = (int)F.size();
      auto mod = T::mod();
      while ((int)inv.size() <= n) {
        int i = inv.size();
        inv.push_back((-inv[mod % i]) * (mod / i));
      }
      F.insert(begin(F), T(0));
      for (int i = 1; i <= n; i++) F[i] *= inv[i];
    };

    auto inplace_diff = [](P &F) -> void {
      if (F.empty()) return;
      F.erase(begin(F));
      T coeff = 1, one = 1;
      for (int i = 0; i < (int)F.size(); i++) {
        F[i] *= coeff;
        coeff += one;
      }
    };

    P b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
    for (int m = 2; m < deg; m *= 2) {
      auto y = b;
      y.resize(2 * m);
      NTT::ntt(y);
      z1 = z2;
      P z(m);
      for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
      NTT::intt(z);
      fill(begin(z), begin(z) + m / 2, T(0));
      NTT::ntt(z);
      for (int i = 0; i < m; ++i) z[i] *= -z1[i];
      NTT::intt(z);
      c.insert(end(c), begin(z) + m / 2, end(z));
      z2 = c;
      z2.resize(2 * m);
      NTT::ntt(z2);
      P x(begin(*this), begin(*this) + min<int>(this->size(), m));
      inplace_diff(x);
      x.push_back(T(0));
      NTT::ntt(x);
      for (int i = 0; i < m; ++i) x[i] *= y[i];
      NTT::intt(x);
      x -= b.diff();
      x.resize(2 * m);
      for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0);
      NTT::ntt(x);
      for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
      NTT::intt(x);
      x.pop_back();
      inplace_integral(x);
      for (int i = m; i < min<int>(this->size(), 2 * m); ++i)
        x[i] += (*this)[i];
      fill(begin(x), begin(x) + m, T(0));
      NTT::ntt(x);
      for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
      NTT::intt(x);
      b.insert(end(b), begin(x) + m, end(x));
    }
    return P{begin(b), begin(b) + deg};
  }

  // https://judge.yosupo.jp/problem/pow_of_formal_power_series
  P pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      P ret(deg, T(0));
      ret[0] = T(1);
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if (i * k > deg) return P(deg, T(0));
      if ((*this)[i] != T(0)) {
        T rev = T(1) / (*this)[i];
        P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, T(0));
        return ret;
      }
    }
    return *this;
  }

  P mod_pow(int64_t k, P g) const {
    P modinv = g.rev().inv();
    auto get_div = [&](P base) {
      if (base.size() < g.size()) {
        base.clear();
        return base;
      }
      int n = base.size() - g.size() + 1;
      return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
    };
    P x(*this), ret{1};
    while (k > 0) {
      if (k & 1) {
        ret *= x;
        ret -= get_div(ret) * g;
        ret.shrink();
      }
      x *= x;
      x -= get_div(x) * g;
      x.shrink();
      k >>= 1;
    }
    return ret;
  }

  // https://judge.yosupo.jp/problem/polynomial_taylor_shift
  P taylor_shift(T c) const {
    int n = (int)this->size();
    vector<T> fact(n), rfact(n);
    fact[0] = rfact[0] = T(1);
    for (int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
    rfact[n - 1] = T(1) / fact[n - 1];
    for (int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
    P p(*this);
    for (int i = 0; i < n; i++) p[i] *= fact[i];
    p = p.rev();
    P bs(n, T(1));
    for (int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
    p = (p * bs).pre(n);
    p = p.rev();
    for (int i = 0; i < n; i++) p[i] *= rfact[i];
    return p;
  }
};

template <typename Mint>
using FPS = FormalPowerSeriesFriendlyNTT<Mint>;

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

#line 1 "math/fft/number-theoretic-transform-friendly-mod-int.hpp"
/**
 * @brief Number Theoretic Transform Friendly ModInt
 */
template <typename Mint>
struct NumberTheoreticTransformFriendlyModInt {
  static vector<Mint> roots, iroots, rate3, irate3;
  static int max_base;

  NumberTheoreticTransformFriendlyModInt() = default;

  static void init() {
    if (roots.empty()) {
      const unsigned mod = Mint::mod();
      assert(mod >= 3 && mod % 2 == 1);
      auto tmp = mod - 1;
      max_base = 0;
      while (tmp % 2 == 0) tmp >>= 1, max_base++;
      Mint root = 2;
      while (root.pow((mod - 1) >> 1) == 1) {
        root += 1;
      }
      assert(root.pow(mod - 1) == 1);

      roots.resize(max_base + 1);
      iroots.resize(max_base + 1);
      rate3.resize(max_base + 1);
      irate3.resize(max_base + 1);

      roots[max_base] = root.pow((mod - 1) >> max_base);
      iroots[max_base] = Mint(1) / roots[max_base];
      for (int i = max_base - 1; i >= 0; i--) {
        roots[i] = roots[i + 1] * roots[i + 1];
        iroots[i] = iroots[i + 1] * iroots[i + 1];
      }
      {
        Mint prod = 1, iprod = 1;
        for (int i = 0; i <= max_base - 3; i++) {
          rate3[i] = roots[i + 3] * prod;
          irate3[i] = iroots[i + 3] * iprod;
          prod *= iroots[i + 3];
          iprod *= roots[i + 3];
        }
      }
    }
  }

  static void ntt(vector<Mint> &a) {
    init();
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    int h = __builtin_ctz(n);
    assert(h <= max_base);
    int len = 0;
    Mint imag = roots[2];
    if (h & 1) {
      int p = 1 << (h - 1);
      Mint rot = 1;
      for (int i = 0; i < p; i++) {
        auto r = a[i + p];
        a[i + p] = a[i] - r;
        a[i] += r;
      }
      len++;
    }
    for (; len + 1 < h; len += 2) {
      int p = 1 << (h - len - 2);
      {  // s = 0
        for (int i = 0; i < p; i++) {
          auto a0 = a[i];
          auto a1 = a[i + p];
          auto a2 = a[i + 2 * p];
          auto a3 = a[i + 3 * p];
          auto a1na3imag = (a1 - a3) * imag;
          auto a0a2 = a0 + a2;
          auto a1a3 = a1 + a3;
          auto a0na2 = a0 - a2;
          a[i] = a0a2 + a1a3;
          a[i + 1 * p] = a0a2 - a1a3;
          a[i + 2 * p] = a0na2 + a1na3imag;
          a[i + 3 * p] = a0na2 - a1na3imag;
        }
      }
      Mint rot = rate3[0];
      for (int s = 1; s < (1 << len); s++) {
        int offset = s << (h - len);
        Mint rot2 = rot * rot;
        Mint rot3 = rot2 * rot;
        for (int i = 0; i < p; i++) {
          auto a0 = a[i + offset];
          auto a1 = a[i + offset + p] * rot;
          auto a2 = a[i + offset + 2 * p] * rot2;
          auto a3 = a[i + offset + 3 * p] * rot3;
          auto a1na3imag = (a1 - a3) * imag;
          auto a0a2 = a0 + a2;
          auto a1a3 = a1 + a3;
          auto a0na2 = a0 - a2;
          a[i + offset] = a0a2 + a1a3;
          a[i + offset + 1 * p] = a0a2 - a1a3;
          a[i + offset + 2 * p] = a0na2 + a1na3imag;
          a[i + offset + 3 * p] = a0na2 - a1na3imag;
        }
        rot *= rate3[__builtin_ctz(~s)];
      }
    }
  }

  static void intt(vector<Mint> &a, bool f = true) {
    init();
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    int h = __builtin_ctz(n);
    assert(h <= max_base);
    int len = h;
    Mint iimag = iroots[2];
    for (; len > 1; len -= 2) {
      int p = 1 << (h - len);
      {  // s = 0
        for (int i = 0; i < p; i++) {
          auto a0 = a[i];
          auto a1 = a[i + 1 * p];
          auto a2 = a[i + 2 * p];
          auto a3 = a[i + 3 * p];
          auto a2na3iimag = (a2 - a3) * iimag;
          auto a0na1 = a0 - a1;
          auto a0a1 = a0 + a1;
          auto a2a3 = a2 + a3;
          a[i] = a0a1 + a2a3;
          a[i + 1 * p] = (a0na1 + a2na3iimag);
          a[i + 2 * p] = (a0a1 - a2a3);
          a[i + 3 * p] = (a0na1 - a2na3iimag);
        }
      }
      Mint irot = irate3[0];
      for (int s = 1; s < (1 << (len - 2)); s++) {
        int offset = s << (h - len + 2);
        Mint irot2 = irot * irot;
        Mint irot3 = irot2 * irot;
        for (int i = 0; i < p; i++) {
          auto a0 = a[i + offset];
          auto a1 = a[i + offset + 1 * p];
          auto a2 = a[i + offset + 2 * p];
          auto a3 = a[i + offset + 3 * p];
          auto a2na3iimag = (a2 - a3) * iimag;
          auto a0na1 = a0 - a1;
          auto a0a1 = a0 + a1;
          auto a2a3 = a2 + a3;
          a[i + offset] = a0a1 + a2a3;
          a[i + offset + 1 * p] = (a0na1 + a2na3iimag) * irot;
          a[i + offset + 2 * p] = (a0a1 - a2a3) * irot2;
          a[i + offset + 3 * p] = (a0na1 - a2na3iimag) * irot3;
        }
        irot *= irate3[__builtin_ctz(~s)];
      }
    }
    if (len >= 1) {
      int p = 1 << (h - 1);
      for (int i = 0; i < p; i++) {
        auto ajp = a[i] - a[i + p];
        a[i] += a[i + p];
        a[i + p] = ajp;
      }
    }
    if (f) {
      Mint inv_sz = Mint(1) / n;
      for (int i = 0; i < n; i++) a[i] *= inv_sz;
    }
  }

  static vector<Mint> multiply(vector<Mint> a, vector<Mint> b) {
    int need = a.size() + b.size() - 1;
    int nbase = 1;
    while ((1 << nbase) < need) nbase++;
    int sz = 1 << nbase;
    a.resize(sz, 0);
    b.resize(sz, 0);
    ntt(a);
    ntt(b);
    Mint inv_sz = Mint(1) / sz;
    for (int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz;
    intt(a, false);
    a.resize(need);
    return a;
  }
};

template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::roots =
    vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::iroots =
    vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate3 =
    vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate3 =
    vector<Mint>();
template <typename Mint>
int NumberTheoreticTransformFriendlyModInt<Mint>::max_base = 0;
#line 4 "math/fps/formal-power-series-friendly-ntt.hpp"

template <typename T>
struct FormalPowerSeriesFriendlyNTT : vector<T> {
  using vector<T>::vector;
  using P = FormalPowerSeriesFriendlyNTT;
  using NTT = NumberTheoreticTransformFriendlyModInt<T>;

  P pre(int deg) const {
    return P(begin(*this), begin(*this) + min((int)this->size(), deg));
  }

  P rev(int deg = -1) const {
    P ret(*this);
    if (deg != -1) ret.resize(deg, T(0));
    reverse(begin(ret), end(ret));
    return ret;
  }

  void shrink() {
    while (this->size() && this->back() == T(0)) this->pop_back();
  }

  P operator+(const P &r) const { return P(*this) += r; }

  P operator+(const T &v) const { return P(*this) += v; }

  P operator-(const P &r) const { return P(*this) -= r; }

  P operator-(const T &v) const { return P(*this) -= v; }

  P operator*(const P &r) const { return P(*this) *= r; }

  P operator*(const T &v) const { return P(*this) *= v; }

  P operator/(const P &r) const { return P(*this) /= r; }

  P operator%(const P &r) const { return P(*this) %= r; }

  P &operator+=(const P &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;
  }

  P &operator-=(const P &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;
  }

  // https://judge.yosupo.jp/problem/convolution_mod
  P &operator*=(const P &r) {
    if (this->empty() || r.empty()) {
      this->clear();
      return *this;
    }
    auto ret = NTT::multiply(*this, r);
    return *this = {begin(ret), end(ret)};
  }

  P &operator/=(const P &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
  }

  P &operator%=(const P &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }

  // https://judge.yosupo.jp/problem/division_of_polynomials
  pair<P, P> div_mod(const P &r) {
    P q = *this / r;
    P x = *this - q * r;
    x.shrink();
    return make_pair(q, x);
  }

  P operator-() const {
    P ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  P &operator+=(const T &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

  P &operator-=(const T &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }

  P &operator*=(const T &v) {
    for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= v;
    return *this;
  }

  P dot(P r) const {
    P ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }

  P operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    P ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }

  P operator<<(int sz) const {
    P ret(*this);
    ret.insert(ret.begin(), sz, T(0));
    return ret;
  }

  T operator()(T x) const {
    T r = 0, w = 1;
    for (auto &v : *this) {
      r += w * v;
      w *= x;
    }
    return r;
  }

  P diff() const {
    const int n = (int)this->size();
    P ret(max(0, n - 1));
    for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
    return ret;
  }

  P integral() const {
    const int n = (int)this->size();
    P ret(n + 1);
    ret[0] = T(0);
    for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
    return ret;
  }

  // https://judge.yosupo.jp/problem/inv_of_formal_power_series
  // F(0) must not be 0
  P inv(int deg = -1) const {
    assert(((*this)[0]) != T(0));
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    P res(deg);
    res[0] = {T(1) / (*this)[0]};
    for (int d = 1; d < deg; d <<= 1) {
      P f(2 * d), g(2 * d);
      for (int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
      for (int j = 0; j < d; j++) g[j] = res[j];
      NTT::ntt(f);
      NTT::ntt(g);
      f = f.dot(g);
      NTT::intt(f);
      for (int j = 0; j < d; j++) f[j] = 0;
      NTT::ntt(f);
      for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
      NTT::intt(f);
      for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
    }
    return res;
  }

  // https://judge.yosupo.jp/problem/log_of_formal_power_series
  // F(0) must be 1
  P log(int deg = -1) const {
    assert((*this)[0] == T(1));
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
  P sqrt(
      int deg = -1,
      const function<T(T)> &get_sqrt = [](T) { return T(1); }) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if ((*this)[0] == T(0)) {
      for (int i = 1; i < n; i++) {
        if ((*this)[i] != T(0)) {
          if (i & 1) return {};
          if (deg - i / 2 <= 0) break;
          auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
          if (ret.empty()) return {};
          ret = ret << (i / 2);
          if ((int)ret.size() < deg) ret.resize(deg, T(0));
          return ret;
        }
      }
      return P(deg, 0);
    }
    auto sqr = T(get_sqrt((*this)[0]));
    if (sqr * sqr != (*this)[0]) return {};
    P ret{sqr};
    T inv2 = T(1) / T(2);
    for (int i = 1; i < deg; i <<= 1) {
      ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
    }
    return ret.pre(deg);
  }

  P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
    return sqrt(deg, get_sqrt);
  }

  // https://judge.yosupo.jp/problem/exp_of_formal_power_series
  // F(0) must be 0
  P exp(int deg = -1) const {
    if (deg == -1) deg = this->size();
    assert((*this)[0] == T(0));

    P inv;
    inv.reserve(deg + 1);
    inv.push_back(T(0));
    inv.push_back(T(1));

    auto inplace_integral = [&](P &F) -> void {
      const int n = (int)F.size();
      auto mod = T::mod();
      while ((int)inv.size() <= n) {
        int i = inv.size();
        inv.push_back((-inv[mod % i]) * (mod / i));
      }
      F.insert(begin(F), T(0));
      for (int i = 1; i <= n; i++) F[i] *= inv[i];
    };

    auto inplace_diff = [](P &F) -> void {
      if (F.empty()) return;
      F.erase(begin(F));
      T coeff = 1, one = 1;
      for (int i = 0; i < (int)F.size(); i++) {
        F[i] *= coeff;
        coeff += one;
      }
    };

    P b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
    for (int m = 2; m < deg; m *= 2) {
      auto y = b;
      y.resize(2 * m);
      NTT::ntt(y);
      z1 = z2;
      P z(m);
      for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
      NTT::intt(z);
      fill(begin(z), begin(z) + m / 2, T(0));
      NTT::ntt(z);
      for (int i = 0; i < m; ++i) z[i] *= -z1[i];
      NTT::intt(z);
      c.insert(end(c), begin(z) + m / 2, end(z));
      z2 = c;
      z2.resize(2 * m);
      NTT::ntt(z2);
      P x(begin(*this), begin(*this) + min<int>(this->size(), m));
      inplace_diff(x);
      x.push_back(T(0));
      NTT::ntt(x);
      for (int i = 0; i < m; ++i) x[i] *= y[i];
      NTT::intt(x);
      x -= b.diff();
      x.resize(2 * m);
      for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0);
      NTT::ntt(x);
      for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
      NTT::intt(x);
      x.pop_back();
      inplace_integral(x);
      for (int i = m; i < min<int>(this->size(), 2 * m); ++i)
        x[i] += (*this)[i];
      fill(begin(x), begin(x) + m, T(0));
      NTT::ntt(x);
      for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
      NTT::intt(x);
      b.insert(end(b), begin(x) + m, end(x));
    }
    return P{begin(b), begin(b) + deg};
  }

  // https://judge.yosupo.jp/problem/pow_of_formal_power_series
  P pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      P ret(deg, T(0));
      ret[0] = T(1);
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if (i * k > deg) return P(deg, T(0));
      if ((*this)[i] != T(0)) {
        T rev = T(1) / (*this)[i];
        P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, T(0));
        return ret;
      }
    }
    return *this;
  }

  P mod_pow(int64_t k, P g) const {
    P modinv = g.rev().inv();
    auto get_div = [&](P base) {
      if (base.size() < g.size()) {
        base.clear();
        return base;
      }
      int n = base.size() - g.size() + 1;
      return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
    };
    P x(*this), ret{1};
    while (k > 0) {
      if (k & 1) {
        ret *= x;
        ret -= get_div(ret) * g;
        ret.shrink();
      }
      x *= x;
      x -= get_div(x) * g;
      x.shrink();
      k >>= 1;
    }
    return ret;
  }

  // https://judge.yosupo.jp/problem/polynomial_taylor_shift
  P taylor_shift(T c) const {
    int n = (int)this->size();
    vector<T> fact(n), rfact(n);
    fact[0] = rfact[0] = T(1);
    for (int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
    rfact[n - 1] = T(1) / fact[n - 1];
    for (int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
    P p(*this);
    for (int i = 0; i < n; i++) p[i] *= fact[i];
    p = p.rev();
    P bs(n, T(1));
    for (int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
    p = (p * bs).pre(n);
    p = p.rev();
    for (int i = 0; i < n; i++) p[i] *= rfact[i];
    return p;
  }
};

template <typename Mint>
using FPS = FormalPowerSeriesFriendlyNTT<Mint>;

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