Tifa's CP Library

:heavy_check_mark: src/test_cpverifier/library-checker/kth_term_of_linearly_recurrent_sequence.p3ntt-s30-ds.test.cpp

Depends on

Code

#define AUTO_GENERATED
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"

#include "../../code/io/ios_container.hpp"
#include "../../code/math/nth_term_lrec.hpp"

constexpr u32 MOD = 998244353;

#include "../../code/math/mint_ds.hpp"
#include "../../code/math/mint_s30.hpp"
#include "../../code/poly/poly3ntt.hpp"

using mint = tifa_libs::math::mint_ds<-1>;
using mint1 = tifa_libs::math::mint_s30<167772161>;
using mint2 = tifa_libs::math::mint_s30<469762049>;
using mint3 = tifa_libs::math::mint_s30<754974721>;
using poly = tifa_libs::math::poly3ntt<mint, mint1, mint2, mint3>;

int main() {
  mint::set_mod(MOD);
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  u32 d;
  u64 k;
  std::cin >> d >> k;
  vec<mint> a(d), c(d + 1);
  std::cin >> a;
  c[0] = 1;
  for (u32 i = 1; i <= d; ++i) {
    std::cin >> c[i];
    c[i] = -c[i];
  }
  std::cout << tifa_libs::math::nth_term_lrec<poly>(k, a, c) << '\n';
  return 0;
}
#line 1 "src/test_cpverifier/library-checker/kth_term_of_linearly_recurrent_sequence.p3ntt-s30-ds.test.cpp"
#define AUTO_GENERATED
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"

#line 1 "src/code/io/ios_container.hpp"



#line 1 "src/code/util/traits.hpp"



#line 1 "src/code/util/util.hpp"



#include <bits/stdc++.h>

template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }

using i8 = int8_t;
using i16 = int16_t;
using i32 = int32_t;
using i64 = int64_t;
using i128 = __int128_t;
using isz = ptrdiff_t;

using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using u128 = __uint128_t;
using usz = size_t;

using f32 = float;
using f64 = double;
using f128 = long double;

template <class T>
using ptt = std::pair<T, T>;
template <class T>
using pt3 = std::tuple<T, T, T>;
template <class T>
using pt4 = std::tuple<T, T, T, T>;

template <class T, usz N>
using arr = std::array<T, N>;
template <class T>
using vec = std::vector<T>;
template <class T>
using vvec = vec<vec<T>>;
template <class T>
using v3ec = vec<vvec<T>>;
template <class U, class T>
using vecp = vec<std::pair<U, T>>;
template <class U, class T>
using vvecp = vvec<std::pair<U, T>>;
template <class T>
using vecpt = vec<ptt<T>>;
template <class T>
using vvecpt = vvec<ptt<T>>;

template <class T, class C = std::less<T>>
using pq = std::priority_queue<T, vec<T>, C>;
template <class T>
using pqg = std::priority_queue<T, vec<T>, std::greater<T>>;

using strn = std::string;
using strnv = std::string_view;

using vecu = vec<u32>;
using vvecu = vvec<u32>;
using v3ecu = v3ec<u32>;
using vecu64 = vec<u64>;
using vecb = vec<bool>;
using vvecb = vvec<bool>;

#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif

using namespace std::literals;

constexpr i8 operator""_i8(unsigned long long x) { return (i8)x; }
constexpr i16 operator""_i16(unsigned long long x) { return (i16)x; }
constexpr i32 operator""_i32(unsigned long long x) { return (i32)x; }
constexpr i64 operator""_i64(unsigned long long x) { return (i64)x; }
constexpr isz operator""_iz(unsigned long long x) { return (isz)x; }

constexpr u8 operator""_u8(unsigned long long x) { return (u8)x; }
constexpr u16 operator""_u16(unsigned long long x) { return (u16)x; }
constexpr u32 operator""_u32(unsigned long long x) { return (u32)x; }
constexpr u64 operator""_u64(unsigned long long x) { return (u64)x; }
constexpr usz operator""_uz(unsigned long long x) { return (usz)x; }

inline const auto fn_0 = [](auto&&...) {};


#line 5 "src/code/util/traits.hpp"

namespace tifa_libs {

template <class T>
concept iterable_c = requires(T v) {
  { v.begin() } -> std::same_as<typename T::iterator>;
  { v.end() } -> std::same_as<typename T::iterator>;
};

template <class T>
concept container_c = iterable_c<T> && !std::derived_from<T, std::basic_string<typename T::value_type>>;

template <class T>
constexpr bool is_char_v = std::is_same_v<T, char> || std::is_same_v<T, signed char> || std::is_same_v<T, unsigned char>;
template <class T>
concept char_c = is_char_v<T>;

template <class T>
constexpr bool is_s128_v = std::is_same_v<T, __int128_t> || std::is_same_v<T, __int128>;
template <class T>
concept s128_c = is_s128_v<T>;

template <class T>
constexpr bool is_u128_v = std::is_same_v<T, __uint128_t> || std::is_same_v<T, unsigned __int128>;
template <class T>
concept u128_c = is_u128_v<T>;

template <class T>
constexpr bool is_i128_v = is_s128_v<T> || is_u128_v<T>;
template <class T>
concept i128_c = is_u128_v<T>;

template <class T>
constexpr bool is_int_v = std::is_integral_v<T> || is_i128_v<T>;
template <class T>
concept int_c = is_int_v<T>;

template <class T>
constexpr bool is_sint_v = is_s128_v<T> || (is_int_v<T> && std::is_signed_v<T>);
template <class T>
concept sint_c = is_sint_v<T>;

template <class T>
constexpr bool is_uint_v = is_u128_v<T> || (is_int_v<T> && std::is_unsigned_v<T>);
template <class T>
concept uint_c = is_uint_v<T>;

template <class T>
concept mint_c = requires(T x) {
  { x.mod() } -> uint_c;
  { x.val() } -> uint_c;
};

template <class T>
constexpr bool is_arithm_v = std::is_arithmetic_v<T> || is_int_v<T>;
template <class T>
concept arithm_c = is_arithm_v<T>;

template <class T>
struct to_sint : std::make_signed<T> {};
template <>
struct to_sint<u128> {
  using type = u128;
};
template <>
struct to_sint<i128> {
  using type = u128;
};
template <class T>
using to_sint_t = typename to_sint<T>::type;

template <class T>
struct to_uint : std::make_unsigned<T> {};
template <>
struct to_uint<u128> {
  using type = u128;
};
template <>
struct to_uint<i128> {
  using type = u128;
};
template <class T>
using to_uint_t = typename to_uint<T>::type;

}  // namespace tifa_libs


#line 5 "src/code/io/ios_container.hpp"

template <tifa_libs::container_c T>
std::istream &operator>>(std::istream &is, T &x) {
  for (auto &i : x) is >> i;
  return is;
}
template <tifa_libs::container_c T>
std::ostream &operator<<(std::ostream &os, T const &x) {
  if (x.begin() == x.end()) return os;
  auto it = x.begin();
  os << *it++;
  for (; it != x.end(); ++it) os << ' ' << *it;
  return os;
}


#line 1 "src/code/math/nth_term_lrec.hpp"



#line 1 "src/code/poly/bostan_mori.hpp"



#line 1 "src/code/conv/ntt_doubling.hpp"



#line 1 "src/code/math/qpow.hpp"



#line 5 "src/code/math/qpow.hpp"

namespace tifa_libs::math {

template <class T>
constexpr T qpow(T a, u64 b, T const& init_v = T{1}) {
  T res = init_v;
  for (; b; b >>= 1, a = a * a)
    if (b & 1) res = res * a;
  return res;
}

}  // namespace tifa_libs::math


#line 1 "src/code/conv/dft_traits.hpp"



#line 5 "src/code/conv/dft_traits.hpp"

namespace tifa_libs {

template <class T>
concept dft_c = requires(T x, vec<typename T::data_t> v, u32 n) {
  { x.size() } -> std::same_as<u32>;
  x.bzr(n);
  x.dif(v, n);
  x.dit(v, n);
};

template <class T>
concept ntt_c = dft_c<T> && requires(T x) {
  T::max_size;
  T::G;
};

}  // namespace tifa_libs


#line 6 "src/code/conv/ntt_doubling.hpp"

namespace tifa_libs::math {

template <ntt_c NTT_t, std::same_as<typename NTT_t::data_t> mint>
constexpr void ntt_doubling(NTT_t const& ntt, vec<mint>& f, u32 n = 0) {
  if (!n) n = (u32)f.size() / 2;
  assert(std::has_single_bit(n) && f.size() >= n * 2);
  vec<mint> g(f.begin(), f.begin() + n);
  ntt.dit(g);
  mint r = 1, zeta = qpow(ntt.G, (mint::mod() - 1) / (n * 2));
  for (u32 i = 0; i < n; ++i) g[i] *= r, r *= zeta;
  ntt.dif(g);
  std::ranges::copy(g, f.begin() + n);
}

}  // namespace tifa_libs::math


#line 1 "src/code/poly/poly.hpp"



#line 5 "src/code/poly/poly.hpp"

namespace tifa_libs::math {

// clang-format off
enum ccore_t { ct_FFT, ct_3NTT, ct_NTT, ct_CNTT };
// clang-format on

template <class mint, class ccore>
requires requires(ccore cc, vec<mint> l, vec<mint> const &r, u32 sz) {
  { ccore::ct_cat } -> std::same_as<ccore_t const &>;
  cc.conv(l, r);
  cc.conv(l, r, sz);
}
class poly {
  vec<mint> d;

 public:
  using value_type = mint;
  using data_type = vec<value_type>;
  using ccore_type = ccore;
  static inline ccore_type conv_core;

  explicit constexpr poly(u32 sz = 1, value_type const &val = value_type{}) : d(sz, val) {}
  constexpr poly(typename data_type::const_iterator begin, typename data_type::const_iterator end) : d(begin, end) {}
  constexpr poly(std::initializer_list<value_type> v) : d(v) {}
  template <class T>
  explicit constexpr poly(vec<T> const &v) : d(v) {}

  friend constexpr std::istream &operator>>(std::istream &is, poly &poly) {
    for (auto &val : poly.d) is >> val;
    return is;
  }
  friend constexpr std::ostream &operator<<(std::ostream &os, poly const &poly) {
    if (!poly.size()) return os;
    for (u32 i = 1; i < poly.size(); ++i) os << poly[i - 1] << ' ';
    return os << poly.d.back();
  }

  constexpr u32 size() const { return (u32)d.size(); }
  constexpr bool empty() const {
    for (auto &&i : d)
      if (i != 0) return 0;
    return 1;
  }
  constexpr data_type &data() { return d; }
  constexpr data_type const &data() const { return d; }

  constexpr value_type &operator[](u32 x) { return d[x]; }
  constexpr value_type const &operator[](u32 x) const { return d[x]; }
  constexpr value_type operator()(value_type x) const {
    value_type ans = 0;
    for (u32 i = size() - 1; ~i; --i) ans = ans * x + d[i];
    return ans;
  }

  template <class F>
  requires requires(F f, u32 idx, mint &val) {
    f(idx, val);
  }
  constexpr void apply_range(u32 l, u32 r, F &&f) {
    assert(l < r && r <= size());
    for (u32 i = l; i < r; ++i) f(i, d[i]);
  }
  template <class F>
  constexpr void apply(F &&f) { apply_range(0, size(), std::forward<F>(f)); }
  constexpr void resize(u32 size) { d.resize(size); }
  constexpr poly pre(u32 size) const {
    poly _ = *this;
    _.resize(size);
    return _;
  }
  constexpr void strip() {
    auto it = std::find_if(d.rbegin(), d.rend(), [](auto const &x) { return x != 0; });
    d.resize(usz(d.rend() - it));
    if (d.empty()) d.push_back(value_type(0));
  }
  friend poly stripped(poly p) {
    p.strip();
    return p;
  }
  constexpr void reverse(u32 n = 0) { std::reverse(d.begin(), d.begin() + (n ? n : size())); }
  constexpr void conv(poly const &r, u32 ans_size = 0) { conv_core.conv(d, r.d, ans_size); }

  constexpr poly operator-() const {
    poly ret = *this;
    ret.apply([](u32, auto &v) { v = -v; });
    return ret;
  }

  friend constexpr poly operator+(poly p, value_type c) {
    p[0] += c;
    return p;
  }
  friend constexpr poly operator+(value_type c, poly const &p) { return p + c; }
  friend constexpr poly operator-(poly p, value_type c) {
    p[0] -= c;
    return p;
  }
  friend constexpr poly operator-(value_type c, poly const &p) { return p - c; }

  constexpr poly &operator*=(value_type c) {
    apply([&c](u32, auto &v) { v *= c; });
    return *this;
  }
  friend constexpr poly operator*(poly p, value_type c) { return p *= c; }
  friend constexpr poly operator*(value_type c, poly p) { return p *= c; }

  constexpr poly &operator+=(poly const &r) {
    if (!r.size()) return *this;
    resize(std::max(size(), r.size()));
    apply_range(0, r.size(), [&r](u32 i, auto &v) { v += r[i]; });
    return *this;
  }
  friend constexpr poly operator+(poly l, poly const &r) { return l += r; }

  constexpr poly &operator-=(poly const &r) {
    if (!r.size()) return *this;
    resize(std::max(size(), r.size()));
    apply_range(0, r.size(), [&r](u32 i, auto &v) { v -= r[i]; });
    return *this;
  }
  friend constexpr poly operator-(poly l, poly const &r) { return l -= r; }

  constexpr poly &operator*=(poly const &r) {
    if (!r.size()) {
      resize(1);
      d[0] = 0;
      return *this;
    }
    conv(r);
    return *this;
  }
  friend constexpr poly operator*(poly l, poly const &r) { return l *= r; }

  constexpr auto operator<=>(poly const &r) const { return stripped(*this).d <=> stripped(r).d; }
  constexpr bool operator==(poly const &r) const { return stripped(*this).d == stripped(r).d; }
};

}  // namespace tifa_libs::math


#line 6 "src/code/poly/bostan_mori.hpp"

namespace tifa_libs::math {
namespace bostan_mori_impl_ {
template <class ccore_t, class T>
vec<T> coeff_(ccore_t const& core, ccore_t const& core2, vec<T>& q, u64 n, u32 d) {
  static u32 len = core.size();
  static vec<T> s(len * 2);
  static constexpr T inv2 = (T::mod() + 1) / 2;
  if (!n) {
    vec<T> res(d);
    T q0 = 0;
    for (u32 i = 0; i < len; ++i) q0 += q[i];
    res.back() = len * q0.inv();
    return res;
  }
  ntt_doubling(core, q, len);
  vec<T> a(len * 2);
  for (u32 i = 0; i < len * 2; ++i) a[i] = q[i] * q[i ^ 1];
  for (u32 i = 0, j = 0; i < len * 2; i += 2, ++j) a[j] = inv2 * (a[i] + a[i + 1]);
  vec<T> w = coeff_(core, core2, a, n / 2, d);
  for (u32 i = 0; i < len * 2; ++i) s[i] = 0;
  for (u32 i = (n & 1) ^ 1, j = 0; j < d; ++j, i += 2) s[i] = w[j];
  core2.dif(s);
  for (u32 i = 0; i < len * 2; ++i) s[i] *= q[i ^ 1];
  core2.dit(s);
  return vec<T>(s.begin() + d, s.begin() + d * 2);
}
}  // namespace bostan_mori_impl_

// @return [x^k]p/q
template <class mint, class ccore>
constexpr auto bostan_mori(u64 n, poly<mint, ccore> const& p, poly<mint, ccore> const& q) {
  assert(p.size() == q.size() - 1 && !p.empty());
  if constexpr (ccore::ct_cat != ct_NTT) {
    auto p_ = p, q_ = q;
    while (n) {
      auto _ = q_;
      for (u32 i = 1; i < _.size(); i += 2) _[i] = -_[i];
      auto s = p_ * _, t = q_ * _;
      for (u32 i = n & 1; i < s.size(); i += 2) p_[i / 2] = s[i];
      for (u32 i = 0; i < t.size(); i += 2) q_[i / 2] = t[i];
      n /= 2;
    }
    return p_[0];
  } else {
    auto& core = poly<mint, ccore>::conv_core;
    auto core2 = core;
    u32 m = (u32)q.size();
    core.bzr(m);
    core2.bzr(core.size() * 2);
    auto q_ = q.data();
    core.dif(q_);
    q_.resize(core2.size());
    auto iq = bostan_mori_impl_::coeff_(core, core2, q_, n, m - 1);
    mint res = 0;
    for (u32 i = 0, e = u32(iq.size() - 1); i <= e; ++i) res += p[i] * iq[e - i];
    return res;
  }
}

}  // namespace tifa_libs::math


#line 1 "src/code/math/berlekamp_massey.hpp"



#line 5 "src/code/math/berlekamp_massey.hpp"

namespace tifa_libs::math {

template <class T>
constexpr vec<T> berlekamp_massey(vec<T> const &a) {
  u32 n = (u32)a.size();
  vec<T> b{1}, c{1};
  b.reserve(n + 1);
  c.reserve(n + 1);
  T y = 1;
  for (u32 k = 1; k <= n; ++k) {
    u32 l = (u32)c.size();
    T x = 0;
    for (u32 i = 0; i < l; ++i) x += c[i] * a[k - l + i];
    b.push_back(0);
    u32 m = (u32)b.size();
    if (x == 0) continue;
    T d_ = x / y;
    if (l < m) {
      auto _ = c;
      c.insert(c.begin(), m - l, 0);
      for (u32 i = 0; i < m; ++i) c[m - 1 - i] -= d_ * b[m - 1 - i];
      b = _;
      y = x;
    } else
      for (u32 i = 0; i < m; ++i) c[l - 1 - i] -= d_ * b[m - 1 - i];
  }
  std::ranges::reverse(c);
  return c;
}

}  // namespace tifa_libs::math


#line 6 "src/code/math/nth_term_lrec.hpp"

namespace tifa_libs::math {

template <class poly, std::same_as<typename poly::value_type> mint>
constexpr mint nth_term_lrec(u64 n, vec<mint> const& a, vec<mint> const& bm) {
  if (n < a.size()) return a[n];
  assert(!bm.empty() && bm[0] == 1);
  poly q(bm);
  q.strip();
  return bostan_mori(n, (poly(a) * q).pre(q.size() - 1), q);
}
template <class poly, std::same_as<typename poly::value_type> mint>
constexpr mint nth_term_lrec(u64 n, vec<mint> const& a) {
  if (n < a.size()) return a[n];
  auto bm = berlekamp_massey(a);
  return nth_term_lrec<poly, mint>(n, a, bm);
}

}  // namespace tifa_libs::math


#line 6 "src/test_cpverifier/library-checker/kth_term_of_linearly_recurrent_sequence.p3ntt-s30-ds.test.cpp"

constexpr u32 MOD = 998244353;

#line 1 "src/code/math/mint_ds.hpp"



#line 1 "src/code/math/mint.hpp"



#line 1 "src/code/nt/inverse.hpp"



#line 1 "src/code/nt/inv_gcd.hpp"



#line 1 "src/code/math/safe_mod.hpp"



#line 5 "src/code/math/safe_mod.hpp"

namespace tifa_libs::math {

template <sint_c T>
constexpr T safe_mod(T x, to_uint_t<T> mod) { return ((x %= (T)mod) < 0 ? x + (T)mod : x); }

}  // namespace tifa_libs::math


#line 1 "src/code/nt/exgcd.hpp"



#line 5 "src/code/nt/exgcd.hpp"

namespace tifa_libs::math {

// @return tuple(g, x, y) s.t. g = gcd(a, b), xa + yb = g, |x| + |y| is the minimal (primary) and x <= y (secondarily)
template <sint_c T>
constexpr auto exgcd(T a, T b) {
  T x1 = 1, x2 = 0, x3 = 0, x4 = 1;
  while (b) {
    T c = a / b;
    std::tie(x1, x2, x3, x4, a, b) = std::make_tuple(x3, x4, x1 - x3 * c, x2 - x4 * c, b, a - b * c);
  }
  return std::make_tuple(to_uint_t<T>(a), x1, x2);
}

}  // namespace tifa_libs::math


#line 6 "src/code/nt/inv_gcd.hpp"

namespace tifa_libs::math {

template <uint_c T>
constexpr ptt<T> inv_gcd(T n, T mod) {
  using U = to_sint_t<T>;
  auto [g, x, y] = exgcd(U(n % mod), (U)mod);
  return {g, safe_mod(x, mod)};
}

}  // namespace tifa_libs::math


#line 5 "src/code/nt/inverse.hpp"

namespace tifa_libs::math {

template <uint_c T, uint_c U>
constexpr U inverse(T n, U mod) {
  auto [g, x] = inv_gcd(U(n % mod), mod);
  assert(g == 1);
  return x;
}

}  // namespace tifa_libs::math


#line 6 "src/code/math/mint.hpp"

namespace tifa_libs::math {

template <class D, uint_c Rt>
class mint {
  constexpr D const &d() const { return static_cast<D const &>(*this); }
  constexpr D &d() { return static_cast<D &>(*this); }

 protected:
  Rt v_{};

 public:
  constexpr mint() {}
  template <int_c T>
  constexpr mint(T v) : v_(D::mod_(v)) {}
  constexpr operator D() { return d(); }

  using raw_type = Rt;
  using sraw_type = to_sint_t<Rt>;
  static constexpr raw_type mod() { return D::mod_(); }
  static constexpr sraw_type smod() { return (sraw_type)D::mod_(); }
  constexpr raw_type val() const { return d().val_(); }
  constexpr sraw_type sval() const { return (sraw_type)d().val_(); }
  constexpr raw_type &data() { return d().data_(); }

  template <int_c T>
  explicit constexpr operator T() const { return (T)val(); }
  constexpr mint &operator+=(mint const &r) { return d().adde_(r.d()); }
  constexpr mint &operator-=(mint const &r) { return d().sube_(r.d()); }
  constexpr mint &operator*=(mint const &r) { return d().mule_(r.d()); }
  constexpr mint &operator/=(mint const &r) { return *this = *this * r.inv(); }
  constexpr mint const &operator+() const { return *this; }
  constexpr mint operator-() const { return d().neg_(); }
  constexpr mint inv() const { return inverse(val(), mod()); }
  friend constexpr mint operator+(mint l, mint const &r) { return l += r; }
  friend constexpr mint operator-(mint l, mint const &r) { return l -= r; }
  friend constexpr mint operator*(mint l, mint const &r) { return l *= r; }
  friend constexpr mint operator/(mint l, mint const &r) { return l /= r; }
  friend constexpr bool operator==(mint const &l, mint const &r) { return l.val() == r.val(); }
  friend constexpr auto operator<=>(mint const &l, mint const &r) { return l.sval() - r.sval(); }
  friend std::istream &operator>>(std::istream &is, mint &x) {
    i64 _;
    is >> _;
    x = mint(_);
    return is;
  }
  friend std::ostream &operator<<(std::ostream &os, mint const &x) { return os << x.val(); }
  friend constexpr mint abs(mint const &x) { return x.val(); }
};

}  // namespace tifa_libs::math


#line 5 "src/code/math/mint_ds.hpp"

namespace tifa_libs::math {

template <i32 ID>
class mint_ds : public mint<mint_ds<ID>, u32> {
  using base = mint<mint_ds<ID>, u32>;
  friend base;

  struct barrett {
    u32 m_;
    u64 im;
    // @param m `1 <= m < 2^31`
    explicit constexpr barrett(u32 m = 998244353) : m_(m), im(-1_u64 / m + 1) {}
    // @return m
    constexpr u32 umod() const { return m_; }
    constexpr u32 mul(u32 a, u32 b) const {
      u64 z = (u64)a * b, x = (u64)(((u128)z * im) >> 64);
      u32 v = (u32)(z - x * m_);
      return v + (m_ <= v ? m_ : 0);
    }
  };

  static inline barrett bt_;

 public:
  static constexpr bool FIXED_MOD = false;
  static constexpr void set_mod(u32 m) {
    assert(1 <= m);
    bt_ = barrett(m);
  }

  constexpr mint_ds() {}
  template <int_c T>
  constexpr mint_ds(T v) { this->v_ = mod_(v); }

 private:
  using raw_t = typename base::raw_type;
  using sraw_t = typename base::sraw_type;
  template <sint_c T>
  static constexpr raw_t mod_(T v) {
    i64 x = i64(v % (i64)mod_());
    return raw_t(x + (x < 0 ? mod_() : 0));
  }
  template <uint_c T>
  static constexpr raw_t mod_(T v) { return raw_t(v % mod_()); }
  static constexpr raw_t mod_() { return bt_.umod(); }
  constexpr raw_t val_() const { return this->v_; }
  constexpr raw_t &data_() { return this->v_; }

  constexpr mint_ds neg_() const { return -(sraw_t)val_(); }
  constexpr mint_ds &adde_(mint_ds const &r) {
    data_() += r.val_();
    if (val_() >= mod_()) data_() -= mod_();
    return *this;
  }
  constexpr mint_ds &sube_(mint_ds const &r) {
    data_() -= r.val_();
    if (val_() >= mod_()) data_() += mod_();
    return *this;
  }
  constexpr mint_ds &mule_(mint_ds const &r) {
    data_() = bt_.mul(val_(), r.val_());
    return *this;
  }
};

}  // namespace tifa_libs::math


#line 1 "src/code/math/mint_s30.hpp"



#line 5 "src/code/math/mint_s30.hpp"

namespace tifa_libs::math {

template <u32 MOD>
class mint_s30 : public mint<mint_s30<MOD>, u32> {
  using base = mint<mint_s30<MOD>, u32>;
  friend base;

  static constexpr u32 MOD2 = MOD << 1;
  static constexpr u32 R = []() {
    u32 t = 2, iv = MOD * (t - MOD * MOD);
    iv *= t - MOD * iv, iv *= t - MOD * iv;
    return iv * (MOD * iv - t);
  }();
  static constexpr u32 R2 = -(u64)(MOD) % MOD;

  static_assert(MOD & 1);
  static_assert(-R * MOD == 1);
  static_assert((MOD >> 30) == 0);
  static_assert(MOD != 1);

  static constexpr u32 reduce(u64 x) { return u32((x + u64((u32)x * R) * MOD) >> 32); }
  static constexpr u32 norm(u32 x) { return x - (MOD & -((MOD - 1 - x) >> 31)); }

 public:
  static constexpr bool FIXED_MOD = true;
  constexpr mint_s30() {}
  template <int_c T>
  constexpr mint_s30(T v) { this->v_ = mod_(v); }

 private:
  using raw_t = typename base::raw_type;
  using sraw_t = typename base::sraw_type;
  template <sint_c T>
  static constexpr raw_t mod_(T v) { return reduce(u64(v % (i32)mod_() + (i32)mod_()) * R2); }
  template <uint_c T>
  static constexpr raw_t mod_(T v) { return reduce(u64(v % mod_()) * R2); }
  static constexpr raw_t mod_() { return MOD; }
  constexpr raw_t val_() const { return norm(reduce(this->v_)); }
  constexpr raw_t &data_() { return this->v_; }

  constexpr mint_s30 neg_() const {
    mint_s30 res;
    res.v_ = (MOD2 & -(this->v_ != 0)) - this->v_;
    return res;
  }
  constexpr mint_s30 &adde_(mint_s30 const &r) {
    this->v_ += r.v_ - MOD2, this->v_ += MOD2 & -(this->v_ >> 31);
    return *this;
  }
  constexpr mint_s30 &sube_(mint_s30 const &r) {
    this->v_ -= r.v_, this->v_ += MOD2 & -(this->v_ >> 31);
    return *this;
  }
  constexpr mint_s30 &mule_(mint_s30 const &r) {
    this->v_ = reduce(u64(this->v_) * r.v_);
    return *this;
  }
};

}  // namespace tifa_libs::math


#line 1 "src/code/poly/poly3ntt.hpp"



#line 1 "src/code/conv/conv_3ntt.hpp"



#line 1 "src/code/math/mul_mod_u.hpp"



#line 5 "src/code/math/mul_mod_u.hpp"

namespace tifa_libs::math {

constexpr u64 mul_mod_u(u64 a, u64 b, u64 mod) {
  if (std::bit_width(a) + std::bit_width(b) <= 64) return a * b % mod;
  else return (u64)((u128)a * b % mod);
}

}  // namespace tifa_libs::math


#line 1 "src/code/conv/conv_dft.hpp"



#line 1 "src/code/conv/conv_naive.hpp"



#line 5 "src/code/conv/conv_naive.hpp"

namespace tifa_libs::math {

template <class U, class T = U>
requires(sizeof(U) <= sizeof(T))
constexpr vec<T> conv_naive(vec<U> const &l, vec<U> const &r, u32 ans_size = 0) {
  if (l.empty() || r.empty()) return {};
  if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
  u32 n = (u32)l.size(), m = (u32)r.size();
  vec<T> ans(ans_size);
  if (n < m)
    for (u32 j = 0; j < m; ++j)
      for (u32 i = 0; i < n; ++i) {
        if (i + j >= ans_size) break;
        ans[i + j] += (T)l[i] * (T)r[j];
      }
  else
    for (u32 i = 0; i < n; ++i)
      for (u32 j = 0; j < m; ++j) {
        if (i + j >= ans_size) break;
        ans[i + j] += (T)l[i] * (T)r[j];
      }
  return ans;
}

}  // namespace tifa_libs::math


#line 6 "src/code/conv/conv_dft.hpp"

namespace tifa_libs::math {

template <dft_c DFT_t, std::same_as<typename DFT_t::data_t> DFT_data_t>
constexpr vec<DFT_data_t> conv_dft(DFT_t &dft, vec<DFT_data_t> l, vec<DFT_data_t> r, u32 ans_size = 0) {
  if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
  if (ans_size < 32) return conv_naive(l, r, ans_size);
  dft.bzr(std::max({(u32)l.size(), (u32)r.size(), std::min(u32(l.size() + r.size() - 1), ans_size)}));
  dft.dif(l);
  dft.dif(r);
  for (u32 i = 0; i < dft.size(); ++i) l[i] *= r[i];
  dft.dit(l);
  l.resize(ans_size);
  return l;
}
template <class DFT_t, class mint, class T = u64>
constexpr vec<mint> conv_dft_u64(DFT_t &dft, vec<T> const &l, vec<T> const &r, u32 ans_size = 0) {
  if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
  vec<mint> l_, r_;
  l_.reserve(l.size());
  r_.reserve(r.size());
  for (auto i : l) l_.push_back(i);
  for (auto i : r) r_.push_back(i);
  return conv_dft(dft, l_, r_, ans_size);
}

}  // namespace tifa_libs::math


#line 1 "src/code/conv/ntt32.hpp"



#line 1 "src/code/bit/lowbit.hpp"



#line 5 "src/code/bit/lowbit.hpp"

namespace tifa_libs::bit {

template <class T>
constexpr T lowbit(T x) { return T(1) << std::countr_zero(x); }

}  // namespace tifa_libs::bit


#line 1 "src/code/nt/is_prime.hpp"



#line 1 "src/code/math/qpow_mod.hpp"



#line 5 "src/code/math/qpow_mod.hpp"

namespace tifa_libs::math {

constexpr u64 qpow_mod(u64 a, u64 b, u64 mod) {
  u64 res(1);
  for (a %= mod; b; b >>= 1, a = mul_mod_u(a, a, mod))
    if (b & 1) res = mul_mod_u(res, a, mod);
  return res;
}

}  // namespace tifa_libs::math


#line 6 "src/code/nt/is_prime.hpp"

namespace tifa_libs::math {

constexpr bool is_prime(u64 n) {
  if (n <= 2) return n == 2;
  if (~n & 1) return false;
  if (n < 8 || n == 61) return true;

  auto f = [n, d = (n - 1) >> std::countr_zero(n - 1)](auto const& bases) -> bool {
    for (u64 i : bases) {
      if (!(i % n)) continue;
      u64 t = d, y = qpow_mod(i, t, n);
      while (t != n - 1 && y != 1 && y != n - 1) {
        y = mul_mod_u(y, y, n);
        t *= 2;
      }
      if (y != n - 1 && (~t & 1)) return false;
    }
    return true;
  };

  if (n < (1 << 30)) {
    constexpr u64 bases[3] = {2, 7, 61};
    return f(bases);
  }
  constexpr u64 bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  return f(bases);
}

}  // namespace tifa_libs::math


#line 1 "src/code/nt/proot_u32.hpp"



#line 1 "src/code/math/isqrt.hpp"



#line 5 "src/code/math/isqrt.hpp"

namespace tifa_libs::math {

constexpr u32 isqrt(u64 x) {
  if (!x) return 0;
  int c = i32(std::bit_width(x) - 1) / 2, sh = 31 - c;
  u32 u = [](u64 x) {
    constexpr u8 TAB[192] = {128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 144, 145, 146, 147, 148, 149, 150, 151, 151, 152, 153, 154, 155, 156, 156, 157, 158, 159, 160, 160, 161, 162, 163, 164, 164, 165, 166, 167, 167, 168, 169, 170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178, 179, 179, 180, 181, 181, 182, 183, 183, 184, 185, 186, 186, 187, 188, 188, 189, 190, 190, 191, 192, 192, 193, 194, 194, 195, 196, 196, 197, 198, 198, 199, 200, 200, 201, 201, 202, 203, 203, 204, 205, 205, 206, 206, 207, 208, 208, 209, 210, 210, 211, 211, 212, 213, 213, 214, 214, 215, 216, 216, 217, 217, 218, 219, 219, 220, 220, 221, 221, 222, 223, 223, 224, 224, 225, 225, 226, 227, 227, 228, 228, 229, 229, 230, 230, 231, 232, 232, 233, 233, 234, 234, 235, 235, 236, 237, 237, 238, 238, 239, 239, 240, 240, 241, 241, 242, 242, 243, 243, 244, 244, 245, 246, 246, 247, 247, 248, 248, 249, 249, 250, 250, 251, 251, 252, 252, 253, 253, 254, 254, 255, 255, 255};
    u32 u = TAB[(x >> 56) - 64];
    u = (u << 7) + (u32)(x >> 41) / u;
    return (u << 15) + (u32)((x >> 17) / u);
  }(x << 2 * sh);
  u >>= sh;
  u -= (u64)u * u > x;
  return u;
}

}  // namespace tifa_libs::math


#line 1 "src/code/nt/is_proot.hpp"



#line 5 "src/code/nt/is_proot.hpp"

namespace tifa_libs::math {

template <std::unsigned_integral T, class It>
constexpr bool is_proot(T g, T m, It pf_begin, It pf_end) {
  if (!g) return false;
  for (; pf_begin != pf_end; ++pf_begin)
    if (qpow_mod(g, (m - 1) / *pf_begin, m) == 1) return false;
  return true;
}

}  // namespace tifa_libs::math


#line 6 "src/code/nt/proot_u32.hpp"

namespace tifa_libs::math {

constexpr u32 proot(u32 m) {
  if (m == 2) return 1;
  if (m == 3 || m == 5) return 2;
  if (m == 104857601 || m == 167772161 || m == 469762049) return 3;
  if (m == 754974721) return 11;
  if (m == 998244353 || m == 1004535809) return 3;
  u32 divs[20] = {2};
  u32 cnt = 1, x = (m - 1) / 2;
  x >>= std::countr_zero(x);
  for (u32 i = 3, ed_ = isqrt(x); i <= ed_; i += 2)
    if (x % i == 0) {
      divs[cnt++] = i;
      while (x % i == 0) x /= i;
    }
  if (x > 1) divs[cnt++] = x;
  for (u32 g = 2;; ++g)
    if (is_proot(g, m, divs, divs + cnt)) return g;
}

}  // namespace tifa_libs::math


#line 8 "src/code/conv/ntt32.hpp"

namespace tifa_libs::math {

template <class mint>
struct NTT32 {
  using data_t = mint;

  static_assert(is_prime(mint::mod()) && sizeof(mint::mod()) <= 4 && (mint::mod() & 3) == 1, "MOD must be prime with 4k+1");
  static constexpr u32 max_size = bit::lowbit(mint::mod() - 1);

  static constexpr mint G = proot(mint::mod());

  explicit constexpr NTT32() : root() {}

  constexpr u32 size() const { return (u32)root.size(); }
  constexpr void bzr(u32 len = max_size) {
    u32 n = std::bit_ceil(len);
    assert(n <= max_size);
    if (n == size()) return;
    root.resize(n);
    root[0] = 1;
    mint w = qpow(G, (mint::mod() - 1) / n);
    for (u32 i = 1; i < n; ++i) root[i] = root[i - 1] * w;
  }

#pragma GCC diagnostic ignored "-Wsign-conversion"
  constexpr void dif(vec<mint> &f, u32 n = 0) const {
    assert(size());
    if (!n) n = size();
    if (f.size() < n) f.resize(n);
    assert(std::has_single_bit(n) && n <= size());
    for (u32 i = n / 2, d = 1; i; i /= 2, d *= 2)
      for (u32 j = 0; j < n; j += i * 2) {
        auto w = root.begin();
        mint u, t;
        for (u32 k = 0; k < i; ++k, w += d) {
          f[j | k] = (u = f[j | k]) + (t = f[i | j | k]);
          f[i | j | k] = (u - t) * (*w);
        }
      }
  }
  constexpr void dit(vec<mint> &f, u32 n = 0) const {
    assert(size());
    if (!n) n = size();
    if (f.size() < n) f.resize(n);
    assert(std::has_single_bit(n) && n <= size());
    for (u32 i = 1, d = n / 2; d; i *= 2, d /= 2)
      for (u32 j = 0; j < n; j += i * 2) {
        auto w = root.begin();
        mint t;
        for (u32 k = 0; k < i; ++k, w += d) {
          f[i | j | k] = f[j | k] - (t = f[i | j | k] * (*w));
          f[j | k] += t;
        }
      }
    std::reverse(f.begin() + 1, f.end());
    mint t = mint(n).inv();
    for (u32 i = 0; i < n; ++i) f[i] *= t;
  }
#pragma GCC diagnostic warning "-Wsign-conversion"

 private:
  vec<mint> root;
};

}  // namespace tifa_libs::math


#line 7 "src/code/conv/conv_3ntt.hpp"

namespace tifa_libs::math {

// 167772161, 469762049, 754974721
template <class mint0, class mint1, class mint2>
constexpr vecu64 conv_3ntt_u64(std::tuple<NTT32<mint0>, NTT32<mint1>, NTT32<mint2>> &ntt3, vecu64 const &l, vecu64 const &r, u64 mod, u32 ans_size = 0) {
  constexpr u32 m0 = mint0::mod(), m1 = mint1::mod(), m2 = mint2::mod();
  const u32 r01 = mint1(m0).inv().val(), r02 = mint2(m0).inv().val(), r12 = mint2(m1).inv().val(), r02r12 = (u32)mul_mod_u(r02, r12, m2);
  const u64 w1 = m0 % mod, w2 = mul_mod_u(m0, m1, mod);

  if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
  auto &[ntt0, ntt1, ntt2] = ntt3;

  vec<mint0> d0 = conv_dft_u64<NTT32<mint0>, mint0>(ntt0, l, r, ans_size);
  vec<mint1> d1 = conv_dft_u64<NTT32<mint1>, mint1>(ntt1, l, r, ans_size);
  vec<mint2> d2 = conv_dft_u64<NTT32<mint2>, mint2>(ntt2, l, r, ans_size);

  vecu64 ret(ans_size);
  for (u32 i = 0; i < ans_size; ++i) {
    u64 n1 = d1[i].val(), n2 = d2[i].val(), a = d0[i].val();
    u64 b = mul_mod_u((n1 + m1 - a), r01, m1), c = mul_mod_u(n2 + m2 - a, r02r12, m2) + mul_mod_u(m2 - b, r12, m2);
    ret[i] = (a + mul_mod_u(b, w1, mod) + mul_mod_u(c % m2, w2, mod)) % mod;
  }
  return ret;
}
template <class mint, class mint0, class mint1, class mint2>
constexpr vec<mint> conv_3ntt(std::tuple<NTT32<mint0>, NTT32<mint1>, NTT32<mint2>> &ntt3, vec<mint> const &l, vec<mint> const &r, u32 ans_size = 0) {
  if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
  if (ans_size < 32) return conv_naive(l, r, ans_size);
  vecu64 l_(l.size()), r_(r.size());
  for (u32 i = 0; i < l.size(); ++i) l_[i] = l[i].val();
  for (u32 i = 0; i < r.size(); ++i) r_[i] = r[i].val();
  vecu64 _ = conv_3ntt_u64(ntt3, l_, r_, mint::mod(), ans_size);
  vec<mint> res(_.size());
  for (u32 i = 0; i < _.size(); ++i) res[i] = _[i];
  return res;
}

}  // namespace tifa_libs::math


#line 6 "src/code/poly/poly3ntt.hpp"

namespace tifa_libs::math {
namespace poly3ntt_impl_ {
template <class mint0, class mint1, class mint2>
struct cconv_3ntt {
  std::tuple<NTT32<mint0>, NTT32<mint1>, NTT32<mint2>> ccore;
  static constexpr auto ct_cat = ct_3NTT;
  template <class mint>
  constexpr void conv(vec<mint> &l, vec<mint> const &r, u32 sz = 0) { l = conv_3ntt(ccore, l, r, sz); }
};
}  // namespace poly3ntt_impl_

template <class mint, class mint0, class mint1, class mint2>
using poly3ntt = poly<mint, poly3ntt_impl_::cconv_3ntt<mint0, mint1, mint2>>;

}  // namespace tifa_libs::math


#line 12 "src/test_cpverifier/library-checker/kth_term_of_linearly_recurrent_sequence.p3ntt-s30-ds.test.cpp"

using mint = tifa_libs::math::mint_ds<-1>;
using mint1 = tifa_libs::math::mint_s30<167772161>;
using mint2 = tifa_libs::math::mint_s30<469762049>;
using mint3 = tifa_libs::math::mint_s30<754974721>;
using poly = tifa_libs::math::poly3ntt<mint, mint1, mint2, mint3>;

int main() {
  mint::set_mod(MOD);
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  u32 d;
  u64 k;
  std::cin >> d >> k;
  vec<mint> a(d), c(d + 1);
  std::cin >> a;
  c[0] = 1;
  for (u32 i = 1; i <= d; ++i) {
    std::cin >> c[i];
    c[i] = -c[i];
  }
  std::cout << tifa_libs::math::nth_term_lrec<poly>(k, a, c) << '\n';
  return 0;
}

Test cases

Env Name Status Elapsed Memory
g++-12 example_00 :heavy_check_mark: AC 9 ms 4 MB
g++-12 max_random_00 :heavy_check_mark: AC 8825 ms 20 MB
g++-12 max_random_01 :heavy_check_mark: AC 8861 ms 20 MB
g++-12 max_random_02 :heavy_check_mark: AC 8845 ms 20 MB
g++-12 near_65536_00 :heavy_check_mark: AC 4244 ms 13 MB
g++-12 near_65536_01 :heavy_check_mark: AC 6300 ms 18 MB
g++-12 near_65536_02 :heavy_check_mark: AC 8541 ms 17 MB
g++-12 random_00 :heavy_check_mark: AC 1820 ms 7 MB
g++-12 random_01 :heavy_check_mark: AC 8408 ms 17 MB
g++-12 random_02 :heavy_check_mark: AC 4132 ms 12 MB
g++-12 small_00 :heavy_check_mark: AC 9 ms 4 MB
g++-12 small_01 :heavy_check_mark: AC 10 ms 4 MB
g++-12 small_02 :heavy_check_mark: AC 10 ms 4 MB
g++-12 small_03 :heavy_check_mark: AC 10 ms 4 MB
g++-12 small_04 :heavy_check_mark: AC 15 ms 4 MB
g++-12 small_05 :heavy_check_mark: AC 9 ms 4 MB
g++-12 small_06 :heavy_check_mark: AC 9 ms 4 MB
g++-12 small_07 :heavy_check_mark: AC 9 ms 4 MB
g++-12 small_08 :heavy_check_mark: AC 9 ms 4 MB
g++-12 small_09 :heavy_check_mark: AC 9 ms 4 MB
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