math/fps/kitamasa.hpp

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• Last update: 2023-03-13 14:46:41+09:00
• Include: #include "math/fps/kitamasa.hpp"

Depends on

• math/convolution/ntt.hpp
• math/fps/fps-template.hpp
• math/fps/fps.hpp
• math/mint.hpp

Verified with

• test/yosupo/Math/Kth-term-of-Linearly-Recurrent-Sequence.test.cpp

Code

#include "fps.hpp"
// #include "fps-arbitrary-mod.hpp"


template <class mint>
mint LinearRecurrence(ll k, FPS<mint> Q, FPS<mint> P){
  Q.shrink();
  mint ret = 0;
  if(P.size() >= Q.size()){
    const auto R = P / Q;
    P -= R * Q;
    P.shrink();
    if(k < (int)R.size()) ret += R[k];
  }
  if((int)P.size() == 0) return ret;
  FPS<mint>::set_fft();
  if(FPS<mint>::ntt_ptr == nullptr){
    P.resize((int)Q.size() - 1);
    while(k){
      auto Q2 = Q;
      for(int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
      const auto S = P * Q2;
      const auto T = Q * Q2;
      if(k & 1){
        for(int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for(int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      }else{
        for(int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for(int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      }
      k >>= 1;
    }
    return ret + P[0];
  }else{
    int N = 1;
    while(N < (int)Q.size()) N <<= 1;
    P.resize(2 * N);
    Q.resize(2 * N);
    P.ntt();
    Q.ntt();
    vector<mint> S(2 * N), T(2 * N);
    vector<int> btr(N);
    for(int i = 0, logn = __builtin_ctz(N); i < (1 << logn); i++){
      btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (logn - 1));
    }
    const mint dw = mint(FPS<mint>::ntt_pr()).inv().pow((mint::get_mod() - 1) / (2 * N));
    while(k){
      mint inv2 = mint(2).inv();
      // even degree of Q(x)Q(-x)

      T.resize(N);
      for(int i = 0; i < N; i++) T[i] = Q[(i << 1) | 0] * Q[(i << 1) | 1];
      S.resize(N);
      if(k & 1){
        // odd degree of P(x)Q(-x)

        for(auto &i : btr) {
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] -
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) * inv2;
          inv2 *= dw;
        }
      }else{
        // even degree of P(x)Q(-x)

        for(int i = 0; i < N; i++){
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] +
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) * inv2;
        }
      }
      swap(P, S);
      swap(Q, T);
      k >>= 1;
      if(k < N) break;
      P.ntt_doubling();
      Q.ntt_doubling();
    }
    P.intt();
    Q.intt();
    return ret + (P * (Q.inv()))[k];
  }
}
template <class mint>
mint kitamasa(ll N, FPS<mint> Q, FPS<mint> a){
  assert(!Q.empty() && Q[0] != 0);
  if(N < (int)a.size()) return a[N];
  assert((int)a.size() >= int(Q.size()) - 1);
  auto P = a.pre((int)Q.size() - 1) * Q;
  P.resize(Q.size() - 1);
  return LinearRecurrence<mint>(N, Q, P);
}

#line 1 "math/convolution/ntt.hpp"
template <class mint>
struct NTT {
  static constexpr uint32_t get_pr(){
    const uint32_t _mod = mint::get_mod();
    using u64 = uint64_t;
    u64 ds[32] = {};
    int idx = 0;
    u64 m = _mod - 1;
    for(u64 i = 2; i * i <= m; i++){
      if(m % i == 0){
        ds[idx++] = i;
        while(m % i == 0) m /= i;
      }
    }
    if(m != 1) ds[idx++] = m;
    uint32_t _pr = 2;
    while(true){
      int flg = 1;
      for(int i = 0; i < idx; i++){
        u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
        while(b){
          if(b & 1) r = r * a % _mod;
          a = a * a % _mod;
          b >>= 1;
        }
        if(r == 1){
          flg = 0;
          break;
        }
      }
      if(flg == 1) break;
      _pr++;
    }
    return _pr;
  };
  static constexpr uint32_t mod = mint::get_mod();
  static constexpr uint32_t pr = get_pr();
  static constexpr int level = __builtin_ctzll(mod - 1);
  mint dw[level], dy[level];
  void setwy(const int k){
    mint w[level], y[level];
    w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
    y[k - 1] = w[k - 1].inv();
    for(int i = k - 2; i > 0; i--)
      w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
    dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
    for(int i = 3; i < k; i++){
      dw[i] = dw[i - 1] * y[i - 2] * w[i];
      dy[i] = dy[i - 1] * w[i - 2] * y[i];
    }
  }
  NTT(){ setwy(level); }
  void fft4(vector<mint> &a, const int k){
    if((int)a.size() <= 1) return;
    if(k == 1){
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      return;
    }
    if(k & 1){
      int v = 1 << (k - 1);
      for(int j = 0; j < v; j++) {
        mint ajv = a[j + v];
        a[j + v] = a[j] - ajv;
        a[j] += ajv;
      }
    }
    int u = 1 << (2 + (k & 1));
    int v = 1 << (k - 2 - (k & 1));
    const mint one = mint(1);
    mint imag = dw[1];
    while(v) {
      // jh = 0

      {
        int j0 = 0;
        int j1 = v;
        int j2 = j1 + v;
        int j3 = j2 + v;
        for(; j0 < v; j0++, j1++, j2++, j3++){
          const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
          const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
          const mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
          a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
          a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
        }
      }
      // jh >= 1

      mint ww = one, xx = one * dw[2], wx = one;
      for(int jh = 4; jh < u;){
        ww = xx * xx, wx = ww * xx;
        int j0 = jh * v;
        int je = j0 + v;
        int j2 = je + v;
        for(; j0 < je; j0++, j2++){
          const mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx;
          const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
          const mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
          a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
          a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
        }
        xx *= dw[__builtin_ctzll((jh += 4))];
      }
      u <<= 2;
      v >>= 2;
    }
  }
  void ifft4(vector<mint> &a, const int k){
    if((int)a.size() <= 1) return;
    if(k == 1){
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      return;
    }
    int u = 1 << (k - 2);
    int v = 1;
    const mint one = mint(1);
    mint imag = dy[1];
    while(u){
      // jh = 0

      {
        int j0 = 0;
        int j1 = v;
        int j2 = v + v;
        int j3 = j2 + v;
        for(; j0 < v; j0++, j1++, j2++, j3++){
          const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
          const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
          const mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
          a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
          a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
        }
      }
      // jh >= 1

      mint ww = one, xx = one * dy[2], yy = one;
      u <<= 2;
      for(int jh = 4; jh < u;){
        ww = xx * xx, yy = xx * imag;
        int j0 = jh * v;
        int je = j0 + v;
        int j2 = je + v;
        for(; j0 < je; j0++, j2++){
          const mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
          const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
          const mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
          a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
          a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
        }
        xx *= dy[__builtin_ctzll(jh += 4)];
      }
      u >>= 4;
      v <<= 2;
    }
    if(k & 1){
      u = 1 << (k - 1);
      for(int j = 0; j < u; j++){
        mint ajv = a[j] - a[j + u];
        a[j] += a[j + u];
        a[j + u] = ajv;
      }
    }
  }
  void ntt(vector<mint> &a){
    if((int)a.size() <= 1) return;
    fft4(a, __builtin_ctz(a.size()));
  }
  void intt(vector<mint> &a){
    if((int)a.size() <= 1) return;
    ifft4(a, __builtin_ctz(a.size()));
    const mint iv = mint(a.size()).inv();
    for(auto &x : a) x *= iv;
  }
  vector<mint> multiply(const vector<mint> &a, const vector<mint> &b){
    const int l = a.size() + b.size() - 1;
    if(min<int>(a.size(), b.size()) <= 40){
      vector<mint> s(l);
      for(int i = 0; i < (int)a.size(); i++)
        for(int j = 0; j < (int)b.size(); j++) s[i + j] += a[i] * b[j];
      return s;
    }
    int k = 2, M = 4;
    while(M < l) M <<= 1, k++;
    setwy(k);
    vector<mint> s(M), t(M);
    for(int i = 0; i < (int)a.size(); i++) s[i] = a[i];
    for(int i = 0; i < (int)b.size(); i++) t[i] = b[i];
    fft4(s, k);
    fft4(t, k);
    for(int i = 0; i < M; i++) s[i] *= t[i];
    ifft4(s, k);
    s.resize(l);
    const mint invm = mint(M).inv();
    for(int i = 0; i < l; i++) s[i] *= invm;
    return s;
  }
  void ntt_doubling(vector<mint> &a){
    const int M = (int)a.size();
    auto b = a;
    intt(b);
    mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
    for(int i = 0; i < M; i++) b[i] *= r, r *= zeta;
    ntt(b);
    copy(begin(b), end(b), back_inserter(a));
  }
};
#line 2 "math/mint.hpp"

template <int mod>
struct Mint {
  ll x;
  constexpr Mint(ll x = 0) : x((x + mod) % mod){}
  static constexpr int get_mod(){ return mod; }
  constexpr Mint operator-() const{ return Mint(-x); }
  constexpr Mint operator+=(const Mint &a){
    if((x += a.x) >= mod) x -= mod;
    return *this;
  }
  constexpr Mint &operator++(){
    if(++x == mod) x = 0;
    return *this;
  }
  constexpr Mint operator++(int){
    Mint temp = *this;
    if(++x == mod) x = 0;
    return temp;
  }
  constexpr Mint &operator-=(const Mint &a){
    if((x -= a.x) < 0) x += mod;
    return *this;
  }
  constexpr Mint &operator--(){
    if(--x < 0) x += mod;
    return *this;
  }
  constexpr Mint operator--(int){
    Mint temp = *this;
    if(--x < 0) x += mod;
    return temp;
  }
  constexpr Mint &operator*=(const Mint &a){
    (x *= a.x) %= mod;
    return *this;
  }
  constexpr Mint operator+(const Mint &a) const{ return Mint(*this) += a; }
  constexpr Mint operator-(const Mint &a) const{ return Mint(*this) -= a; }
  constexpr Mint operator*(const Mint &a) const{ return Mint(*this) *= a; }
  constexpr Mint pow(ll t) const{
    if(!t) return 1;
    Mint res = 1, v = *this;
    while(t){
      if(t & 1) res *= v;
      v *= v;
      t >>= 1;
    }
    return res;
  }
  constexpr Mint inv() const{ return pow(mod - 2); }
  constexpr Mint &operator/=(const Mint &a){ return (*this) *= a.inv(); }
  constexpr Mint operator/(const Mint &a) const{ return Mint(*this) /= a; }
  constexpr bool operator==(const Mint &a) const{ return x == a.x; }
  constexpr bool operator!=(const Mint &a) const{ return x != a.x; }
  constexpr bool operator<(const Mint &a) const{ return x < a.x; }
  constexpr bool operator>(const Mint &a) const{ return x > a.x; }
  friend istream &operator>>(istream &is, Mint &a){ return is >> a.x; }
  friend ostream &operator<<(ostream &os, const Mint &a){ return os << a.x; }
};
//using mint = Mint<1000000007>;
#line 3 "math/fps/fps-template.hpp"

template <class mint>
struct FPS : vector<mint> {
  using vector<mint>::vector;
  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;
    }
    const int n = this->size() - r.size() + 1;
    if((int)r.size() <= 64){
      FPS f(*this), g(r);
      g.shrink();
      const mint coeff = g.back().inv();
      for(auto &x : g) x *= coeff;
      const int deg = (int)f.size() - (int)g.size() + 1;
      const 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) const{
    if((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }
  FPS operator<<(int sz) const{
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }
  FPS diff() const{
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for(int i = 1; i < n; i++){
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }
  FPS integral() const{
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if(n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for(int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for(int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }
  mint eval(mint x) const{
    mint r = 0, w = 1;
    for(auto &v : *this) r += w * v, w *= x;
    return r;
  }
  FPS log(int deg = -1) const{
    assert((*this)[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)){
        const mint rev = mint(1) / (*this)[i];
        FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
        ret *= (*this)[i].pow(k);
        ret = (ret << (i * k)).pre(deg);
        if((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
      if(__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
    }
    return FPS(deg, mint(0));
  }
  static void *ntt_ptr;
  static void set_fft();
  FPS &operator*=(const FPS &r);
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_pr();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <class mint>
void *FPS<mint>::ntt_ptr = nullptr;
#line 2 "math/fps/fps.hpp"

template <class mint>
void FPS<mint>::set_fft(){ if(!ntt_ptr) ntt_ptr = new NTT<mint>; }
template <class mint>
FPS<mint> &FPS<mint>::operator*=(const FPS<mint> &r){
  if(this->empty() || r.empty()){
    this->clear();
    return *this;
  }
  set_fft();
  const auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
  return *this = FPS<mint>(ret.begin(), ret.end());
}
template <class mint>
void FPS<mint>::ntt(){
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
template <class mint>
void FPS<mint>::intt(){
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
template <class mint>
void FPS<mint>::ntt_doubling(){
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
template <class mint>
int FPS<mint>::ntt_pr(){
  set_fft();
  return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}
template <class mint>
FPS<mint> FPS<mint>::inv(int deg) const{
  assert((*this)[0] != mint(0));
  if(deg == -1) deg = (int)this->size();
  FPS<mint> res(deg);
  res[0] = { mint(1) / (*this)[0] };
  for(int d = 1; d < deg; d <<= 1){
    FPS<mint> f(2 * d), g(2 * d);
    for(int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
    for(int j = 0; j < d; j++) g[j] = res[j];
    f.ntt();
    g.ntt();
    for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
    f.intt();
    for(int j = 0; j < d; j++) f[j] = 0;
    f.ntt();
    for(int j = 0; j < 2 * d; j++) f[j] *= g[j];
    f.intt();
    for(int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
  }
  return res.pre(deg);
}
template <class mint>
FPS<mint> FPS<mint>::exp(int deg) const{
  using fps = FPS<mint>;
  assert((*this).size() == 0 || (*this)[0] == mint(0));
  if(deg == -1) deg = this->size();
  fps inv;
  inv.reserve(deg + 1);
  inv.push_back(mint(0));
  inv.push_back(mint(1));
  auto inplace_integral = [&](fps &F) -> void {
    const int n = (int)F.size();
    const auto mod = mint::get_mod();
    while((int)inv.size() <= n){
      const int i = inv.size();
      inv.push_back((-inv[mod % i]) * (mod / i));
    }
    F.insert(begin(F), mint(0));
    for(int i = 1; i <= n; i++) F[i] *= inv[i];
  };
  auto inplace_diff = [](fps& F) -> void {
    if(F.empty()) return;
    F.erase(begin(F));
    mint coeff = 1;
    const mint one = 1;
    for(int i = 0; i < (int)F.size(); i++){
      F[i] *= coeff;
      coeff += one;
    }
  };
  fps 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);
    y.ntt();
    z1 = z2;
    fps z(m);
    for(int i = 0; i < m; i++) z[i] = y[i] * z1[i];
    z.intt();
    fill(begin(z), begin(z) + m / 2, mint(0));
    z.ntt();
    for(int i = 0; i < m; i++) z[i] *= -z1[i];
    z.intt();
    c.insert(end(c), begin(z) + m / 2, end(z));
    z2 = c;
    z2.resize(2 * m);
    z2.ntt();
    fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
    x.resize(m);
    inplace_diff(x);
    x.push_back(mint(0));
    x.ntt();
    for(int i = 0; i < m; i++) x[i] *= y[i];
    x.intt();
    x -= b.diff();
    x.resize(2 * m);
    for(int i = 0; i < m - 1; i++) x[m + i] = x[i], x[i] = mint(0);
    x.ntt();
    for(int i = 0; i < 2 * m; i++) x[i] *= z2[i];
    x.intt();
    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, mint(0));
    x.ntt();
    for(int i = 0; i < 2 * m; i++) x[i] *= y[i];
    x.intt();
    b.insert(end(b), begin(x) + m, end(x));
  }
  return fps{ begin(b), begin(b) + deg };
}
#line 2 "math/fps/kitamasa.hpp"
// #include "fps-arbitrary-mod.hpp"


template <class mint>
mint LinearRecurrence(ll k, FPS<mint> Q, FPS<mint> P){
  Q.shrink();
  mint ret = 0;
  if(P.size() >= Q.size()){
    const auto R = P / Q;
    P -= R * Q;
    P.shrink();
    if(k < (int)R.size()) ret += R[k];
  }
  if((int)P.size() == 0) return ret;
  FPS<mint>::set_fft();
  if(FPS<mint>::ntt_ptr == nullptr){
    P.resize((int)Q.size() - 1);
    while(k){
      auto Q2 = Q;
      for(int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
      const auto S = P * Q2;
      const auto T = Q * Q2;
      if(k & 1){
        for(int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for(int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      }else{
        for(int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for(int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      }
      k >>= 1;
    }
    return ret + P[0];
  }else{
    int N = 1;
    while(N < (int)Q.size()) N <<= 1;
    P.resize(2 * N);
    Q.resize(2 * N);
    P.ntt();
    Q.ntt();
    vector<mint> S(2 * N), T(2 * N);
    vector<int> btr(N);
    for(int i = 0, logn = __builtin_ctz(N); i < (1 << logn); i++){
      btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (logn - 1));
    }
    const mint dw = mint(FPS<mint>::ntt_pr()).inv().pow((mint::get_mod() - 1) / (2 * N));
    while(k){
      mint inv2 = mint(2).inv();
      // even degree of Q(x)Q(-x)

      T.resize(N);
      for(int i = 0; i < N; i++) T[i] = Q[(i << 1) | 0] * Q[(i << 1) | 1];
      S.resize(N);
      if(k & 1){
        // odd degree of P(x)Q(-x)

        for(auto &i : btr) {
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] -
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) * inv2;
          inv2 *= dw;
        }
      }else{
        // even degree of P(x)Q(-x)

        for(int i = 0; i < N; i++){
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] +
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) * inv2;
        }
      }
      swap(P, S);
      swap(Q, T);
      k >>= 1;
      if(k < N) break;
      P.ntt_doubling();
      Q.ntt_doubling();
    }
    P.intt();
    Q.intt();
    return ret + (P * (Q.inv()))[k];
  }
}
template <class mint>
mint kitamasa(ll N, FPS<mint> Q, FPS<mint> a){
  assert(!Q.empty() && Q[0] != 0);
  if(N < (int)a.size()) return a[N];
  assert((int)a.size() >= int(Q.size()) - 1);
  auto P = a.pre((int)Q.size() - 1) * Q;
  P.resize(Q.size() - 1);
  return LinearRecurrence<mint>(N, Q, P);
}

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