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View the Project on GitHub suisen-cp/cp-library-cpp
#define PROBLEM "https://atcoder.jp/contests/abc208/tasks/abc208_f" #include <iostream> #include <atcoder/modint> #include "library/polynomial/lagrange_interpolation.hpp" #include "library/sequence/powers.hpp" using mint = atcoder::modint1000000007; int main() { long long n; int m, k; std::cin >> n >> m >> k; std::vector<mint> f = suisen::powers<mint>(k + m, k); for (int loop = 0; loop < m; ++loop) { for (int i = 1; i <= k + m; ++i) f[i] += f[i - 1]; } std::cout << suisen::lagrange_interpolation_arithmetic_progression<mint>(f, n).val() << '\n'; return 0; }
#line 1 "test/src/polynomial/lagrange_interpolation/cumulative_sum.test.cpp" #define PROBLEM "https://atcoder.jp/contests/abc208/tasks/abc208_f" #include <iostream> #include <atcoder/modint> #line 1 "library/polynomial/lagrange_interpolation.hpp" #line 1 "library/math/product_of_differences.hpp" #include <deque> #line 1 "library/polynomial/multi_point_eval.hpp" #include <vector> namespace suisen { template <typename FPSType, typename T> std::vector<typename FPSType::value_type> multi_point_eval(const FPSType& f, const std::vector<T>& xs) { int n = xs.size(); if (n == 0) return {}; std::vector<FPSType> seg(2 * n); for (int i = 0; i < n; ++i) seg[n + i] = FPSType{ -xs[i], 1 }; for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1]; seg[1] = f % seg[1]; for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i]; std::vector<typename FPSType::value_type> ys(n); for (int i = 0; i < n; ++i) ys[i] = seg[n + i].size() ? seg[n + i][0] : 0; return ys; } } // namespace suisen #line 6 "library/math/product_of_differences.hpp" namespace suisen { /** * O(N(logN)^2) * return the vector p of length xs.size() s.t. p[i]=Π[j!=i](x[i]-x[j]) */ template <typename FPSType, typename T> std::vector<typename FPSType::value_type> product_of_differences(const std::vector<T>& xs) { // f(x):=Π_i(x-x[i]) // => f'(x)=Σ_i Π[j!=i](x-x[j]) // => f'(x[i])=Π[j!=i](x[i]-x[j]) const int n = xs.size(); std::deque<FPSType> dq; for (int i = 0; i < n; ++i) dq.push_back(FPSType{ -xs[i], 1 }); while (dq.size() >= 2) { auto f = std::move(dq.front()); dq.pop_front(); auto g = std::move(dq.front()); dq.pop_front(); dq.push_back(f * g); } auto f = std::move(dq.front()); f.diff_inplace(); return multi_point_eval<FPSType, T>(f, xs); } } // namespace suisen #line 5 "library/polynomial/lagrange_interpolation.hpp" namespace suisen { // O(N^2+NlogP) template <typename T> T lagrange_interpolation_naive(const std::vector<T>& xs, const std::vector<T>& ys, const T t) { const int n = xs.size(); assert(int(ys.size()) == n); T p{ 1 }; for (int i = 0; i < n; ++i) p *= t - xs[i]; T res{ 0 }; for (int i = 0; i < n; ++i) { T w = 1; for (int j = 0; j < n; ++j) if (j != i) w *= xs[i] - xs[j]; res += ys[i] * (t == xs[i] ? 1 : p / (w * (t - xs[i]))); } return res; } // O(N(logN)^2+NlogP) template <typename FPSType, typename T> typename FPSType::value_type lagrange_interpolation(const std::vector<T>& xs, const std::vector<T>& ys, const T t) { const int n = xs.size(); assert(int(ys.size()) == n); std::vector<FPSType> seg(2 * n); for (int i = 0; i < n; ++i) seg[n + i] = FPSType {-xs[i], 1}; for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1]; seg[1] = seg[1].diff() % seg[1]; for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i]; using mint = typename FPSType::value_type; mint p{ 1 }; for (int i = 0; i < n; ++i) p *= t - xs[i]; mint res{ 0 }; for (int i = 0; i < n; ++i) { mint w = seg[n + i][0]; res += ys[i] * (t == xs[i] ? 1 : p / (w * (t - xs[i]))); } return res; } // xs[i] = ai + b // requirement: for all 0≤i<j<n, ai+b ≢ aj+b mod p template <typename T> T lagrange_interpolation_arithmetic_progression(T a, T b, const std::vector<T>& ys, const T t) { const int n = ys.size(); T fac = 1; for (int i = 1; i < n; ++i) fac *= i; std::vector<T> fac_inv(n), suf(n); fac_inv[n - 1] = T(1) / fac; suf[n - 1] = 1; for (int i = n - 1; i > 0; --i) { fac_inv[i - 1] = fac_inv[i] * i; suf[i - 1] = suf[i] * (t - (a * i + b)); } T pre = 1, res = 0; for (int i = 0; i < n; ++i) { T val = ys[i] * pre * suf[i] * fac_inv[i] * fac_inv[n - i - 1]; if ((n - 1 - i) & 1) res -= val; else res += val; pre *= t - (a * i + b); } return res / a.pow(n - 1); } // x = 0, 1, ... template <typename T> T lagrange_interpolation_arithmetic_progression(const std::vector<T>& ys, const T t) { return lagrange_interpolation_arithmetic_progression(T{1}, T{0}, ys, t); } } // namespace suisen #line 1 "library/sequence/powers.hpp" #include <cstdint> #line 1 "library/number/linear_sieve.hpp" #include <cassert> #include <numeric> #line 7 "library/number/linear_sieve.hpp" namespace suisen { // referece: https://37zigen.com/linear-sieve/ class LinearSieve { public: LinearSieve(const int n) : _n(n), min_prime_factor(std::vector<int>(n + 1)) { std::iota(min_prime_factor.begin(), min_prime_factor.end(), 0); prime_list.reserve(_n / 20); for (int d = 2; d <= _n; ++d) { if (min_prime_factor[d] == d) prime_list.push_back(d); const int prime_max = std::min(min_prime_factor[d], _n / d); for (int prime : prime_list) { if (prime > prime_max) break; min_prime_factor[prime * d] = prime; } } } int prime_num() const noexcept { return prime_list.size(); } /** * Returns a vector of primes in [0, n]. * It is guaranteed that the returned vector is sorted in ascending order. */ const std::vector<int>& get_prime_list() const noexcept { return prime_list; } const std::vector<int>& get_min_prime_factor() const noexcept { return min_prime_factor; } /** * Returns a vector of `{ prime, index }`. * It is guaranteed that the returned vector is sorted in ascending order. */ std::vector<std::pair<int, int>> factorize(int n) const noexcept { assert(0 < n and n <= _n); std::vector<std::pair<int, int>> prime_powers; while (n > 1) { int p = min_prime_factor[n], c = 0; do { n /= p, ++c; } while (n % p == 0); prime_powers.emplace_back(p, c); } return prime_powers; } private: const int _n; std::vector<int> min_prime_factor; std::vector<int> prime_list; }; } // namespace suisen #line 6 "library/sequence/powers.hpp" namespace suisen { // returns { 0^k, 1^k, ..., n^k } template <typename mint> std::vector<mint> powers(uint32_t n, uint64_t k) { const auto mpf = LinearSieve(n).get_min_prime_factor(); std::vector<mint> res(n + 1); res[0] = k == 0; for (uint32_t i = 1; i <= n; ++i) res[i] = i == 1 ? 1 : uint32_t(mpf[i]) == i ? mint(i).pow(k) : res[mpf[i]] * res[i / mpf[i]]; return res; } } // namespace suisen #line 8 "test/src/polynomial/lagrange_interpolation/cumulative_sum.test.cpp" using mint = atcoder::modint1000000007; int main() { long long n; int m, k; std::cin >> n >> m >> k; std::vector<mint> f = suisen::powers<mint>(k + m, k); for (int loop = 0; loop < m; ++loop) { for (int i = 1; i <= k + m; ++i) f[i] += f[i - 1]; } std::cout << suisen::lagrange_interpolation_arithmetic_progression<mint>(f, n).val() << '\n'; return 0; }