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#define PROBLEM \ "https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial_limit" #include "../../math/sum_of_exp_times_poly.hpp" #include "../../modint/modint.hpp" #include "../../template/template.hpp" namespace ebi { using mint = modint998244353; void main_() { i64 r, d; std::cin >> r >> d; std::cout << sum_of_exp2_limit<mint>(r, d) << '\n'; } } // namespace ebi int main() { ebi::fast_io(); int t = 1; // std::cin >> t; while (t--) { ebi::main_(); } return 0; }
#line 1 "test/math/Sum_of_Exponential_Times_Polynomial_Limit.test.cpp" #define PROBLEM \ "https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial_limit" #line 2 "math/sum_of_exp_times_poly.hpp" #include <cassert> #include <vector> #line 2 "math/binomial.hpp" #include <bit> #line 5 "math/binomial.hpp" #include <cstdint> #include <iostream> #include <ranges> #line 9 "math/binomial.hpp" #line 2 "modint/base.hpp" #include <concepts> #line 5 "modint/base.hpp" #include <utility> namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 11 "math/binomial.hpp" namespace ebi { template <Modint mint> struct Binomial { private: static void extend(int len = -1) { int sz = (int)fact.size(); if (len < 0) len = 2 * sz; else if (len <= sz) return; else len = std::max(2 * sz, (int)std::bit_ceil(std::uint32_t(len))); len = std::min(len, mint::mod()); assert(sz <= len); fact.resize(len); inv_fact.resize(len); for (int i : std::views::iota(sz, len)) { fact[i] = fact[i - 1] * i; } inv_fact[len - 1] = fact[len - 1].inv(); for (int i : std::views::iota(sz, len) | std::views::reverse) { inv_fact[i - 1] = inv_fact[i] * i; } } public: Binomial() = default; Binomial(int n) { extend(n + 1); } static mint f(int n) { if (n >= (int)fact.size()) [[unlikely]] { extend(n + 1); } return fact[n]; } static mint inv_f(int n) { if (n >= (int)fact.size()) [[unlikely]] { extend(n + 1); } return inv_fact[n]; } static mint c(int n, int r) { if (r < 0 || n < r) return 0; return f(n) * inv_f(r) * inv_f(n - r); } static mint neg_c(int k, int d) { assert(d > 0); return c(k + d - 1, d - 1); } static mint p(int n, int r) { if (r < 0 || n < r) return 0; return f(n) * inv_f(n - r); } static mint catalan_number(int n) { return c(2 * n, n) * inv(n + 1); } static mint inv(int n) { return inv_f(n) * f(n - 1); } static void reserve(int n) { extend(n + 1); } private: static std::vector<mint> fact, inv_fact; }; template <Modint mint> std::vector<mint> Binomial<mint>::fact = std::vector<mint>(2, 1); template <Modint mint> std::vector<mint> Binomial<mint>::inv_fact = std::vector<mint>(2, 1); } // namespace ebi #line 2 "math/lagrange_interpolation.hpp" #line 4 "math/lagrange_interpolation.hpp" /* reference: https://atcoder.jp/contests/abc208/editorial/2195 verify: https://atcoder.jp/contests/abc208/tasks/abc208_f */ namespace ebi { template <class mint> mint lagrange_interpolation(const std::vector<mint> &f, long long n) { const int d = int(f.size()) - 1; // Nのd次以下の多項式 mint fact = 1; std::vector<mint> inv_fact(d + 1); for (int i = 1; i < d + 1; ++i) { fact *= i; } inv_fact[d] = fact.inv(); for (int i = d; i > 0; i--) { inv_fact[i - 1] = inv_fact[i] * i; } std::vector<mint> l(d + 1), r(d + 1); l[0] = 1; for (int i = 0; i < d; ++i) { l[i + 1] = l[i] * (n - i); } r[d] = 1; for (int i = d; i > 0; --i) { r[i - 1] = r[i] * (n - i); } mint res = 0; for (int i = 0; i < d + 1; ++i) { res += mint((d - i) % 2 == 1 ? -1 : 1) * f[i] * l[i] * r[i] * inv_fact[i] * inv_fact[d - i]; } return res; } } // namespace ebi #line 2 "math/linear_sieve.hpp" #line 2 "template/int_alias.hpp" #line 4 "template/int_alias.hpp" namespace ebi { using ld = long double; using std::size_t; using i8 = std::int8_t; using u8 = std::uint8_t; using i16 = std::int16_t; using u16 = std::uint16_t; using i32 = std::int32_t; using u32 = std::uint32_t; using i64 = std::int64_t; using u64 = std::uint64_t; using i128 = __int128_t; using u128 = __uint128_t; } // namespace ebi #line 5 "math/linear_sieve.hpp" /* reference: https://37zigen.com/linear-sieve/ verify: https://atcoder.jp/contests/abc162/submissions/25095562 */ #line 13 "math/linear_sieve.hpp" namespace ebi { struct linear_sieve { private: using u64 = std::uint64_t; int n; std::vector<int> sieve; std::vector<int> prime; public: linear_sieve(int _n) : n(_n), sieve(std::vector<int>(_n + 1, -1)) { for (int i = 2; i <= n; i++) { if (sieve[i] < 0) { sieve[i] = i; prime.emplace_back(i); } for (auto p : prime) { if (u64(p) * u64(i) > u64(n) || p > sieve[i]) break; sieve[p * i] = p; } } } std::vector<int> prime_table() const { return prime; } std::vector<std::pair<int, int>> prime_power_table(int m) const { assert(m <= n); std::vector<std::pair<int, int>> table(m + 1, {1, 1}); for (int i = 2; i <= m; i++) { int p = sieve[i]; table[i] = {p, p}; if (sieve[i / p] == p) { table[i] = table[i / p]; table[i].second *= p; } } return table; } std::vector<std::pair<int, int>> factorize(int x) { assert(x <= n); std::vector<std::pair<int, int>> res; while (x > 1) { int p = sieve[x]; int exp = 0; if (p < 0) { res.emplace_back(x, 1); break; } while (sieve[x] == p) { x /= p; exp++; } res.emplace_back(p, exp); } return res; } std::vector<int> divisors(int x) { assert(x <= n); std::vector<int> res; res.emplace_back(1); auto pf = factorize(x); for (auto p : pf) { int sz = (int)res.size(); for (int i = 0; i < sz; i++) { int ret = 1; for (int j = 0; j < p.second; j++) { ret *= p.first; res.emplace_back(res[i] * ret); } } } return res; } template <class T> std::vector<T> fast_zeta(const std::vector<T> &f) { std::vector<T> F = f; int sz = f.size(); assert(sz <= n + 1); for (int i = 2; i < sz; i++) { if (sieve[i] != i) continue; for (int j = (sz - 1) / i; j >= 1; j--) { F[j] += F[j * i]; } } return F; } template <class T> std::vector<T> fast_mobius(const std::vector<T> &F) { std::vector<T> f = F; int sz = F.size(); assert(sz <= n + 1); for (int i = 2; i < sz; i++) { if (sieve[i] != i) continue; for (int j = 1; j * i < sz; j++) { f[j] -= f[j * i]; } } return f; } template <Modint mint> std::vector<mint> pow_table(int m, int k) { assert(m <= n && k >= 0); std::vector<mint> table(m + 1, 1); table[0] = (k == 0); for (int i = 2; i <= m; i++) { if (sieve[i] == i) { table[i] = mint(i).pow(k); continue; } table[i] = table[sieve[i]] * table[i / sieve[i]]; } return table; } template <Modint mint> std::vector<mint> inv_table() { return pow_table(mint::mod() - 2); } }; } // namespace ebi #line 11 "math/sum_of_exp_times_poly.hpp" namespace ebi { template <Modint mint> mint sum_of_exp_times_poly(const std::vector<mint> &f, mint a, i64 n) { if (n == 0) return 0; if (a == 0) return f[0]; if (a == 1) { std::vector<mint> g(f.size() + 1, 0); for (int i = 1; i < (int)g.size(); i++) { g[i] = g[i - 1] + f[i - 1]; } return lagrange_interpolation(g, n); } int k = (int)f.size() - 1; Binomial<mint> binom(k + 1); std::vector<mint> g(k + 1, 0); { mint pow_a = 1; for (int i = 0; i < k + 1; i++) { g[i] = f[i] * pow_a; pow_a *= a; } for (int i = 0; i < k; i++) { g[i + 1] += g[i]; } } mint c = 0; { mint pow_neg_a = 1; for (int i = 0; i < k + 1; i++) { c += binom.c(k + 1, i) * g[k - i] * pow_neg_a; pow_neg_a *= -a; } } c /= (1 - a).pow(k + 1); { mint inv_a_pow = 1, inv_a = a.inv(); for (int i = 0; i < k + 1; i++) { g[i] = (-c + g[i]) * inv_a_pow; inv_a_pow *= inv_a; } } mint tn = lagrange_interpolation(g, n - 1); return tn * a.pow(n - 1) + c; } template <Modint mint> mint sum_of_exp_times_poly_limit(const std::vector<mint> &f, mint a) { assert(a != 1); if (a == 0) return f[0]; int k = (int)f.size() - 1; Binomial<mint> binom(k + 1); std::vector<mint> g(k + 1, 0); { mint pow_a = 1; for (int i = 0; i < k + 1; i++) { g[i] = f[i] * pow_a; pow_a *= a; } for (int i = 0; i < k; i++) { g[i + 1] += g[i]; } } mint c = 0; { mint pow_neg_a = 1; for (int i = 0; i < k + 1; i++) { c += binom.c(k + 1, i) * g[k - i] * pow_neg_a; pow_neg_a *= -a; } } c /= (1 - a).pow(k + 1); return c; } template <Modint mint> mint sum_of_exp2(mint r, int d, i64 n) { linear_sieve sieve(d); auto f = sieve.pow_table<mint>(d, d); return sum_of_exp_times_poly(f, r, n); } template <Modint mint> mint sum_of_exp2_limit(mint r, int d) { linear_sieve sieve(d); auto f = sieve.pow_table<mint>(d, d); return sum_of_exp_times_poly_limit(f, r); } } // namespace ebi #line 2 "modint/modint.hpp" #line 5 "modint/modint.hpp" #line 7 "modint/modint.hpp" namespace ebi { template <int m> struct static_modint { private: using modint = static_modint; public: static constexpr int mod() { return m; } static constexpr modint raw(int v) { modint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} template <std::signed_integral T> constexpr static_modint(T v) { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template <std::unsigned_integral T> constexpr static_modint(T v) { _v = (unsigned int)(v % umod()); } constexpr unsigned int val() const { return _v; } constexpr unsigned int value() const { return val(); } constexpr modint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } constexpr modint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } constexpr modint operator++(int) { modint res = *this; ++*this; return res; } constexpr modint operator--(int) { modint res = *this; --*this; return res; } constexpr modint &operator+=(const modint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr modint &operator-=(const modint &rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr modint &operator*=(const modint &rhs) { unsigned long long x = _v; x *= rhs._v; _v = (unsigned int)(x % (unsigned long long)umod()); return *this; } constexpr modint &operator/=(const modint &rhs) { return *this = *this * rhs.inv(); } constexpr modint operator+() const { return *this; } constexpr modint operator-() const { return modint() - *this; } constexpr modint pow(long long n) const { assert(0 <= n); modint x = *this, res = 1; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } constexpr modint inv() const { assert(_v); return pow(umod() - 2); } friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += rhs; } friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= rhs; } friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= rhs; } friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.val() == rhs.val(); } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } private: unsigned int _v = 0; static constexpr unsigned int umod() { return m; } }; using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; } // namespace ebi #line 1 "template/template.hpp" #include <bits/stdc++.h> #define rep(i, a, n) for (int i = (int)(a); i < (int)(n); i++) #define rrep(i, a, n) for (int i = ((int)(n)-1); i >= (int)(a); i--) #define Rep(i, a, n) for (i64 i = (i64)(a); i < (i64)(n); i++) #define RRep(i, a, n) for (i64 i = ((i64)(n)-i64(1)); i >= (i64)(a); i--) #define all(v) (v).begin(), (v).end() #define rall(v) (v).rbegin(), (v).rend() #line 2 "template/debug_template.hpp" #line 4 "template/debug_template.hpp" namespace ebi { #ifdef LOCAL #define debug(...) \ std::cerr << "LINE: " << __LINE__ << " [" << #__VA_ARGS__ << "]:", \ debug_out(__VA_ARGS__) #else #define debug(...) #endif void debug_out() { std::cerr << std::endl; } template <typename Head, typename... Tail> void debug_out(Head h, Tail... t) { std::cerr << " " << h; if (sizeof...(t) > 0) std::cerr << " :"; debug_out(t...); } } // namespace ebi #line 2 "template/io.hpp" #line 5 "template/io.hpp" #include <optional> #line 7 "template/io.hpp" namespace ebi { template <typename T1, typename T2> std::ostream &operator<<(std::ostream &os, const std::pair<T1, T2> &pa) { return os << pa.first << " " << pa.second; } template <typename T1, typename T2> std::istream &operator>>(std::istream &os, std::pair<T1, T2> &pa) { return os >> pa.first >> pa.second; } template <typename T> std::ostream &operator<<(std::ostream &os, const std::vector<T> &vec) { for (std::size_t i = 0; i < vec.size(); i++) os << vec[i] << (i + 1 == vec.size() ? "" : " "); return os; } template <typename T> std::istream &operator>>(std::istream &os, std::vector<T> &vec) { for (T &e : vec) std::cin >> e; return os; } template <typename T> std::ostream &operator<<(std::ostream &os, const std::optional<T> &opt) { if (opt) { os << opt.value(); } else { os << "invalid value"; } return os; } void fast_io() { std::cout << std::fixed << std::setprecision(15); std::cin.tie(nullptr); std::ios::sync_with_stdio(false); } } // namespace ebi #line 2 "template/utility.hpp" #line 5 "template/utility.hpp" #line 2 "graph/base.hpp" #line 7 "graph/base.hpp" #line 2 "data_structure/simple_csr.hpp" #line 6 "data_structure/simple_csr.hpp" namespace ebi { template <class E> struct simple_csr { simple_csr() = default; simple_csr(int n, const std::vector<std::pair<int, E>>& elements) : start(n + 1, 0), elist(elements.size()) { for (auto e : elements) { start[e.first + 1]++; } for (auto i : std::views::iota(0, n)) { start[i + 1] += start[i]; } auto counter = start; for (auto [i, e] : elements) { elist[counter[i]++] = e; } } simple_csr(const std::vector<std::vector<E>>& es) : start(es.size() + 1, 0) { int n = es.size(); for (auto i : std::views::iota(0, n)) { start[i + 1] = (int)es[i].size() + start[i]; } elist.resize(start.back()); for (auto i : std::views::iota(0, n)) { std::copy(es[i].begin(), es[i].end(), elist.begin() + start[i]); } } int size() const { return (int)start.size() - 1; } const auto operator[](int i) const { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } auto operator[](int i) { return std::ranges::subrange(elist.begin() + start[i], elist.begin() + start[i + 1]); } const auto operator()(int i, int l, int r) const { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } auto operator()(int i, int l, int r) { return std::ranges::subrange(elist.begin() + start[i] + l, elist.begin() + start[i + 1] + r); } private: std::vector<int> start; std::vector<E> elist; }; } // namespace ebi #line 9 "graph/base.hpp" namespace ebi { template <class T> struct Edge { int from, to; T cost; int id; }; template <class E> struct Graph { using cost_type = E; using edge_type = Edge<cost_type>; Graph(int n_) : n(n_) {} Graph() = default; void add_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); edges.emplace_back(edge_type{u, v, c, m++}); } void add_undirected_edge(int u, int v, cost_type c) { buff.emplace_back(u, edge_type{u, v, c, m}); buff.emplace_back(v, edge_type{v, u, c, m}); edges.emplace_back(edge_type{u, v, c, m}); m++; } void read_tree(int offset = 1, bool is_weighted = false) { read_graph(n - 1, offset, false, is_weighted); } void read_parents(int offset = 1) { for (auto i : std::views::iota(1, n)) { int p; std::cin >> p; p -= offset; add_undirected_edge(p, i, 1); } build(); } void read_graph(int e, int offset = 1, bool is_directed = false, bool is_weighted = false) { for (int i = 0; i < e; i++) { int u, v; std::cin >> u >> v; u -= offset; v -= offset; if (is_weighted) { cost_type c; std::cin >> c; if (is_directed) { add_edge(u, v, c); } else { add_undirected_edge(u, v, c); } } else { if (is_directed) { add_edge(u, v, 1); } else { add_undirected_edge(u, v, 1); } } } build(); } void build() { assert(!prepared); csr = simple_csr<edge_type>(n, buff); buff.clear(); prepared = true; } int size() const { return n; } int node_number() const { return n; } int edge_number() const { return m; } edge_type get_edge(int i) const { return edges[i]; } std::vector<edge_type> get_edges() const { return edges; } const auto operator[](int i) const { return csr[i]; } auto operator[](int i) { return csr[i]; } private: int n, m = 0; std::vector<std::pair<int,edge_type>> buff; std::vector<edge_type> edges; simple_csr<edge_type> csr; bool prepared = false; }; } // namespace ebi #line 8 "template/utility.hpp" namespace ebi { template <class T> inline bool chmin(T &a, T b) { if (a > b) { a = b; return true; } return false; } template <class T> inline bool chmax(T &a, T b) { if (a < b) { a = b; return true; } return false; } template <class T> T safe_ceil(T a, T b) { if (a % b == 0) return a / b; else if (a >= 0) return (a / b) + 1; else return -((-a) / b); } template <class T> T safe_floor(T a, T b) { if (a % b == 0) return a / b; else if (a >= 0) return a / b; else return -((-a) / b) - 1; } constexpr i64 LNF = std::numeric_limits<i64>::max() / 4; constexpr int INF = std::numeric_limits<int>::max() / 2; const std::vector<int> dy = {1, 0, -1, 0, 1, 1, -1, -1}; const std::vector<int> dx = {0, 1, 0, -1, 1, -1, 1, -1}; } // namespace ebi #line 7 "test/math/Sum_of_Exponential_Times_Polynomial_Limit.test.cpp" namespace ebi { using mint = modint998244353; void main_() { i64 r, d; std::cin >> r >> d; std::cout << sum_of_exp2_limit<mint>(r, d) << '\n'; } } // namespace ebi int main() { ebi::fast_io(); int t = 1; // std::cin >> t; while (t--) { ebi::main_(); } return 0; }