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#define PROBLEM \ "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence" #include "../../convolution/ntt.hpp" #include "../../fps/bostan_mori.hpp" #include "../../template/template.hpp" #include "../../utility/modint.hpp" using namespace lib; using mint = modint998244353; int main() { int d; long long k; std::cin >> d >> k; std::vector<mint> a(d), c(d); for (int i = 0; i < d; i++) { int val; std::cin >> val; a[i] = val; } for (int i = 0; i < d; i++) { int val; std::cin >> val; c[i] = val; } std::cout << kitamasa<mint, convolution>(k, a, c).val() << '\n'; }
#line 1 "test/polynomial/Kth_term_of_Linearly_Recurrent_Sequence.test.cpp" #define PROBLEM \ "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence" #line 2 "convolution/ntt.hpp" #line 2 "template/template.hpp" #include <bits/stdc++.h> #define rep(i, s, n) for (int i = (int)(s); i < (int)(n); i++) #define rrep(i, s, n) for (int i = (int)(n)-1; i >= (int)(s); i--) #define all(v) v.begin(), v.end() using ll = long long; using ld = long double; using ull = unsigned long long; template <typename T> bool chmin(T &a, const T &b) { if (a <= b) return false; a = b; return true; } template <typename T> bool chmax(T &a, const T &b) { if (a >= b) return false; a = b; return true; } namespace lib { using namespace std; } // namespace lib // using namespace lib; #line 2 "utility/modint.hpp" #line 4 "utility/modint.hpp" namespace lib { template <ll m> struct modint { using mint = modint; ll a; modint(ll x = 0) : a((x % m + m) % m) {} static constexpr ll mod() { return m; } ll val() const { return a; } ll& val() { return a; } mint pow(ll n) const { mint res = 1; mint x = a; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } mint inv() const { return pow(m - 2); } mint& operator+=(const mint rhs) { a += rhs.a; if (a >= m) a -= m; return *this; } mint& operator-=(const mint rhs) { if (a < rhs.a) a += m; a -= rhs.a; return *this; } mint& operator*=(const mint rhs) { a = a * rhs.a % m; return *this; } mint& operator/=(mint rhs) { *this *= rhs.inv(); return *this; } 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 modint &lhs, const modint &rhs) { return lhs.a == rhs.a; } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } }; using modint998244353 = modint<998244353>; using modint1000000007 = modint<1'000'000'007>; } // namespace lib #line 5 "convolution/ntt.hpp" namespace lib { using mint = modint998244353; struct ntt_info { static constexpr int rank2 = 23; const int g = 3; std::array<std::array<mint, rank2 + 1>, 2> root; ntt_info() { root[0][rank2] = mint(g).pow((mint::mod() - 1) >> rank2); root[1][rank2] = root[0][rank2].inv(); rrep(i, 0, rank2) { root[0][i] = root[0][i + 1] * root[0][i + 1]; root[1][i] = root[1][i + 1] * root[1][i + 1]; } } }; void butterfly(std::vector<mint>& a, bool inverse) { static ntt_info info; int n = a.size(); int bit_size = 0; while ((1 << bit_size) < n) bit_size++; assert(1 << bit_size == n); for (int i = 0, j = 1; j < n - 1; j++) { for (int k = n >> 1; k > (i ^= k); k >>= 1); if (j < i) { std::swap(a[i], a[j]); } } rep(bit, 0, bit_size) { rep(i, 0, n / (1 << (bit + 1))) { mint zeta1 = 1; mint zeta2 = info.root[inverse][1]; mint w = info.root[inverse][bit + 1]; rep(j, 0, 1 << bit) { int idx = i * (1 << (bit + 1)) + j; int jdx = idx + (1 << bit); mint p1 = a[idx]; mint p2 = a[jdx]; a[idx] = p1 + zeta1 * p2; a[jdx] = p1 + zeta2 * p2; zeta1 *= w; zeta2 *= w; } } } if (inverse) { mint inv_n = mint(n).inv(); rep(i, 0, n) a[i] *= inv_n; } } std::vector<mint> convolution(const std::vector<mint>& f, const std::vector<mint>& g) { int n = 1; while (n < int(f.size() + g.size() - 1)) n <<= 1; std::vector<mint> a(n), b(n); std::copy(f.begin(), f.end(), a.begin()); std::copy(g.begin(), g.end(), b.begin()); butterfly(a, false); butterfly(b, false); rep(i, 0, n) { a[i] *= b[i]; } butterfly(a, true); a.resize(f.size() + g.size() - 1); return a; } } // namespace lib #line 2 "fps/bostan_mori.hpp" #line 4 "fps/bostan_mori.hpp" namespace lib { template <class T, std::vector<T> (*convolution)(const std::vector<T> &, const std::vector<T> &)> T bostan_mori_algorithm(long long n, std::vector<T> p, std::vector<T> q) { while (n > 0) { auto q_neg = q; for (int i = 1; i < (int)q_neg.size(); i += 2) q_neg[i] = -q_neg[i]; p = convolution(p, q_neg); q = convolution(q, q_neg); for (int i = (n & 1LL); i < (int)p.size(); i += 2) p[i >> 1] = p[i]; p.resize(((int)p.size() + 1 - (n & 1LL)) / 2); for (int i = 0; i < (int)q.size(); i += 2) q[i >> 1] = q[i]; q.resize(((int)q.size() + 1) / 2); n >>= 1; } return p[0] / q[0]; } template <class T, std::vector<T> (*convolution)(const std::vector<T> &, const std::vector<T> &)> T kitamasa(std::int64_t n, std::vector<T> a, std::vector<T> c) { if (n < (int)a.size()) return a[n]; const int d = c.size(); for (auto &val : c) val = -val; c.insert(c.begin(), 1); auto p = convolution(a, c); p.resize(d); return bostan_mori_algorithm<T, convolution>(n, p, c); } } // namespace lib #line 8 "test/polynomial/Kth_term_of_Linearly_Recurrent_Sequence.test.cpp" using namespace lib; using mint = modint998244353; int main() { int d; long long k; std::cin >> d >> k; std::vector<mint> a(d), c(d); for (int i = 0; i < d; i++) { int val; std::cin >> val; a[i] = val; } for (int i = 0; i < d; i++) { int val; std::cin >> val; c[i] = val; } std::cout << kitamasa<mint, convolution>(k, a, c).val() << '\n'; }