C++ Library for Competitive Programming
View the Project on GitHub emthrm/cp-library
/* * @title 数学/形式的冪級数/ファウルハーバーの公式 形式的冪級数版 * * verification-helper: PROBLEM https://yukicoder.me/problems/no/665 */ #include <iostream> #include <vector> #include "emthrm/math/convolution/mod_convolution.hpp" #include "emthrm/math/formal_power_series/faulhaber_by_fps.hpp" #include "emthrm/math/formal_power_series/formal_power_series.hpp" #include "emthrm/math/modint.hpp" int main() { constexpr int MOD = 1000000007; using ModInt = emthrm::MInt<MOD>; emthrm::FormalPowerSeries<ModInt>::set_mult( [](const std::vector<ModInt>& a, const std::vector<ModInt>& b) -> std::vector<ModInt> { return emthrm::mod_convolution(a, b); }); long long n; int k; std::cin >> n >> k; std::cout << emthrm::faulhaber_by_fps<MOD>(n + 1, k) << '\n'; return 0; }
#line 1 "test/math/formal_power_series/faulhaber_by_fps.test.cpp" /* * @title 数学/形式的冪級数/ファウルハーバーの公式 形式的冪級数版 * * verification-helper: PROBLEM https://yukicoder.me/problems/no/665 */ #include <iostream> #include <vector> #line 1 "include/emthrm/math/convolution/mod_convolution.hpp" #include <algorithm> #include <bit> #include <cmath> #line 8 "include/emthrm/math/convolution/mod_convolution.hpp" #line 1 "include/emthrm/math/convolution/fast_fourier_transform.hpp" #line 6 "include/emthrm/math/convolution/fast_fourier_transform.hpp" #include <cassert> #line 8 "include/emthrm/math/convolution/fast_fourier_transform.hpp" #include <iterator> #include <utility> #line 11 "include/emthrm/math/convolution/fast_fourier_transform.hpp" namespace emthrm { namespace fast_fourier_transform { using Real = double; struct Complex { Real re, im; explicit Complex(const Real re = 0, const Real im = 0) : re(re), im(im) {} inline Complex operator+(const Complex& x) const { return Complex(re + x.re, im + x.im); } inline Complex operator-(const Complex& x) const { return Complex(re - x.re, im - x.im); } inline Complex operator*(const Complex& x) const { return Complex(re * x.re - im * x.im, re * x.im + im * x.re); } inline Complex mul_real(const Real r) const { return Complex(re * r, im * r); } inline Complex mul_pin(const Real r) const { return Complex(-im * r, re * r); } inline Complex conj() const { return Complex(re, -im); } }; std::vector<int> butterfly{0}; std::vector<std::vector<Complex>> zeta{{Complex(1, 0)}}; void init(const int n) { const int prev_n = butterfly.size(); if (n <= prev_n) return; butterfly.resize(n); const int prev_lg = zeta.size(); const int lg = std::countr_zero(static_cast<unsigned int>(n)); for (int i = 1; i < prev_n; ++i) { butterfly[i] <<= lg - prev_lg; } for (int i = prev_n; i < n; ++i) { butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1)); } zeta.resize(lg); for (int i = prev_lg; i < lg; ++i) { zeta[i].resize(1 << i); const Real angle = -3.14159265358979323846 * 2 / (1 << (i + 1)); for (int j = 0; j < (1 << (i - 1)); ++j) { zeta[i][j << 1] = zeta[i - 1][j]; const Real theta = angle * ((j << 1) + 1); zeta[i][(j << 1) + 1] = Complex(std::cos(theta), std::sin(theta)); } } } void dft(std::vector<Complex>* a) { assert(std::has_single_bit(a->size())); const int n = a->size(); init(n); const int shift = std::countr_zero(butterfly.size()) - std::countr_zero(a->size()); for (int i = 0; i < n; ++i) { const int j = butterfly[i] >> shift; if (i < j) std::swap((*a)[i], (*a)[j]); } for (int block = 1, den = 0; block < n; block <<= 1, ++den) { for (int i = 0; i < n; i += (block << 1)) { for (int j = 0; j < block; ++j) { const Complex tmp = (*a)[i + j + block] * zeta[den][j]; (*a)[i + j + block] = (*a)[i + j] - tmp; (*a)[i + j] = (*a)[i + j] + tmp; } } } } template <typename T> std::vector<Complex> real_dft(const std::vector<T>& a) { const int n = a.size(); std::vector<Complex> c(std::bit_ceil(a.size())); for (int i = 0; i < n; ++i) { c[i].re = a[i]; } dft(&c); return c; } void idft(std::vector<Complex>* a) { const int n = a->size(); dft(a); std::reverse(std::next(a->begin()), a->end()); const Real r = 1. / n; std::transform(a->begin(), a->end(), a->begin(), [r](const Complex& c) -> Complex { return c.mul_real(r); }); } template <typename T> std::vector<Real> convolution(const std::vector<T>& a, const std::vector<T>& b) { const int a_size = a.size(), b_size = b.size(), c_size = a_size + b_size - 1; const int n = std::max(std::bit_ceil(static_cast<unsigned int>(c_size)), 2U); const int hlf = n >> 1, qtr = hlf >> 1; std::vector<Complex> c(n); for (int i = 0; i < a_size; ++i) { c[i].re = a[i]; } for (int i = 0; i < b_size; ++i) { c[i].im = b[i]; } dft(&c); c.front() = Complex(c.front().re * c.front().im, 0); for (int i = 1; i < hlf; ++i) { const Complex i_square = c[i] * c[i], j_square = c[n - i] * c[n - i]; c[i] = (j_square.conj() - i_square).mul_pin(0.25); c[n - i] = (i_square.conj() - j_square).mul_pin(0.25); } c[hlf] = Complex(c[hlf].re * c[hlf].im, 0); c.front() = (c.front() + c[hlf] + (c.front() - c[hlf]).mul_pin(1)).mul_real(0.5); const int den = std::countr_zero(static_cast<unsigned int>(hlf)); for (int i = 1; i < qtr; ++i) { const int j = hlf - i; const Complex tmp1 = c[i] + c[j].conj(); const Complex tmp2 = ((c[i] - c[j].conj()) * zeta[den][j]).mul_pin(1); c[i] = (tmp1 - tmp2).mul_real(0.5); c[j] = (tmp1 + tmp2).mul_real(0.5).conj(); } if (qtr > 0) c[qtr] = c[qtr].conj(); c.resize(hlf); idft(&c); std::vector<Real> res(c_size); for (int i = 0; i < c_size; i += 2) { res[i] = c[i >> 1].re; } for (int i = 1; i < c_size; i += 2) { res[i] = c[i >> 1].im; } return res; } } // namespace fast_fourier_transform } // namespace emthrm #line 1 "include/emthrm/math/modint.hpp" #ifndef ARBITRARY_MODINT #line 6 "include/emthrm/math/modint.hpp" #endif #include <compare> #line 9 "include/emthrm/math/modint.hpp" // #include <numeric> #line 12 "include/emthrm/math/modint.hpp" namespace emthrm { #ifndef ARBITRARY_MODINT template <unsigned int M> struct MInt { unsigned int v; constexpr MInt() : v(0) {} constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static constexpr int get_mod() { return M; } static constexpr void set_mod(const int divisor) { assert(std::cmp_equal(divisor, M)); } static void init(const int x) { inv<true>(x); fact(x); fact_inv(x); } template <bool MEMOIZES = false> static MInt inv(const int n) { // assert(0 <= n && n < M && std::gcd(n, M) == 1); static std::vector<MInt> inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[M % i] * raw(M / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = M; b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector<MInt> factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector<MInt> f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv<true>(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } constexpr MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } constexpr MInt& operator+=(const MInt& x) { if ((v += x.v) >= M) v -= M; return *this; } constexpr MInt& operator-=(const MInt& x) { if ((v += M - x.v) >= M) v -= M; return *this; } constexpr MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % M; return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } constexpr auto operator<=>(const MInt& x) const = default; constexpr MInt& operator++() { if (++v == M) [[unlikely]] v = 0; return *this; } constexpr MInt operator++(int) { const MInt res = *this; ++*this; return res; } constexpr MInt& operator--() { v = (v == 0 ? M - 1 : v - 1); return *this; } constexpr MInt operator--(int) { const MInt res = *this; --*this; return res; } constexpr MInt operator+() const { return *this; } constexpr MInt operator-() const { return raw(v ? M - v : 0); } constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; } constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; } constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } }; #else // ARBITRARY_MODINT template <int ID> struct MInt { unsigned int v; constexpr MInt() : v(0) {} MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static int get_mod() { return mod(); } static void set_mod(const unsigned int divisor) { mod() = divisor; } static void init(const int x) { inv<true>(x); fact(x); fact_inv(x); } template <bool MEMOIZES = false> static MInt inv(const int n) { // assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1); static std::vector<MInt> inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[mod() % i] * raw(mod() / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = mod(); b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector<MInt> factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector<MInt> f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv<true>(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } MInt& operator+=(const MInt& x) { if ((v += x.v) >= mod()) v -= mod(); return *this; } MInt& operator-=(const MInt& x) { if ((v += mod() - x.v) >= mod()) v -= mod(); return *this; } MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % mod(); return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } auto operator<=>(const MInt& x) const = default; MInt& operator++() { if (++v == mod()) [[unlikely]] v = 0; return *this; } MInt operator++(int) { const MInt res = *this; ++*this; return res; } MInt& operator--() { v = (v == 0 ? mod() - 1 : v - 1); return *this; } MInt operator--(int) { const MInt res = *this; --*this; return res; } MInt operator+() const { return *this; } MInt operator-() const { return raw(v ? mod() - v : 0); } MInt operator+(const MInt& x) const { return MInt(*this) += x; } MInt operator-(const MInt& x) const { return MInt(*this) -= x; } MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } private: static unsigned int& mod() { static unsigned int divisor = 0; return divisor; } }; #endif // ARBITRARY_MODINT } // namespace emthrm #line 11 "include/emthrm/math/convolution/mod_convolution.hpp" namespace emthrm { template <int PRECISION = 15, unsigned int T> std::vector<MInt<T>> mod_convolution(const std::vector<MInt<T>>& a, const std::vector<MInt<T>>& b) { using ModInt = MInt<T>; const int a_size = a.size(), b_size = b.size(), c_size = a_size + b_size - 1; const int n = std::max(std::bit_ceil(static_cast<unsigned int>(c_size)), 2U); constexpr int mask = (1 << PRECISION) - 1; std::vector<fast_fourier_transform::Complex> x(n), y(n); std::transform( a.begin(), a.end(), x.begin(), [mask](const MInt<T>& x) -> fast_fourier_transform::Complex { return fast_fourier_transform::Complex(x.v & mask, x.v >> PRECISION); }); fast_fourier_transform::dft(&x); std::transform( b.begin(), b.end(), y.begin(), [mask](const MInt<T>& y) -> fast_fourier_transform::Complex { return fast_fourier_transform::Complex(y.v & mask, y.v >> PRECISION); }); fast_fourier_transform::dft(&y); const int half = n >> 1; fast_fourier_transform::Complex tmp_a = x.front(), tmp_b = y.front(); x.front() = fast_fourier_transform::Complex(tmp_a.re * tmp_b.re, tmp_a.im * tmp_b.im); y.front() = fast_fourier_transform::Complex( tmp_a.re * tmp_b.im + tmp_a.im * tmp_b.re, 0); for (int i = 1; i < half; ++i) { const int j = n - i; const fast_fourier_transform::Complex a_l_i = (x[i] + x[j].conj()).mul_real(0.5); const fast_fourier_transform::Complex a_h_i = (x[j].conj() - x[i]).mul_pin(0.5); const fast_fourier_transform::Complex b_l_i = (y[i] + y[j].conj()).mul_real(0.5); const fast_fourier_transform::Complex b_h_i = (y[j].conj() - y[i]).mul_pin(0.5); const fast_fourier_transform::Complex a_l_j = (x[j] + x[i].conj()).mul_real(0.5); const fast_fourier_transform::Complex a_h_j = (x[i].conj() - x[j]).mul_pin(0.5); const fast_fourier_transform::Complex b_l_j = (y[j] + y[i].conj()).mul_real(0.5); const fast_fourier_transform::Complex b_h_j = (y[i].conj() - y[j]).mul_pin(0.5); x[i] = a_l_i * b_l_i + (a_h_i * b_h_i).mul_pin(1); y[i] = a_l_i * b_h_i + a_h_i * b_l_i; x[j] = a_l_j * b_l_j + (a_h_j * b_h_j).mul_pin(1); y[j] = a_l_j * b_h_j + a_h_j * b_l_j; } tmp_a = x[half]; tmp_b = y[half]; x[half] = fast_fourier_transform::Complex( tmp_a.re * tmp_b.re, tmp_a.im * tmp_b.im); y[half] = fast_fourier_transform::Complex( tmp_a.re * tmp_b.im + tmp_a.im * tmp_b.re, 0); fast_fourier_transform::idft(&x); fast_fourier_transform::idft(&y); std::vector<ModInt> res(c_size); const ModInt tmp1 = 1 << PRECISION, tmp2 = 1LL << (PRECISION << 1); for (int i = 0; i < c_size; ++i) { res[i] = tmp1 * std::llround(y[i].re) + tmp2 * std::llround(x[i].im) + std::llround(x[i].re); } return res; } } // namespace emthrm #line 1 "include/emthrm/math/formal_power_series/faulhaber_by_fps.hpp" #line 6 "include/emthrm/math/formal_power_series/faulhaber_by_fps.hpp" #line 1 "include/emthrm/math/formal_power_series/bernoulli_number.hpp" #line 5 "include/emthrm/math/formal_power_series/bernoulli_number.hpp" #line 1 "include/emthrm/math/formal_power_series/formal_power_series.hpp" #line 6 "include/emthrm/math/formal_power_series/formal_power_series.hpp" #include <functional> #include <initializer_list> #line 9 "include/emthrm/math/formal_power_series/formal_power_series.hpp" #include <numeric> #line 11 "include/emthrm/math/formal_power_series/formal_power_series.hpp" namespace emthrm { template <typename T> struct FormalPowerSeries { std::vector<T> coef; explicit FormalPowerSeries(const int deg = 0) : coef(deg + 1, 0) {} explicit FormalPowerSeries(const std::vector<T>& coef) : coef(coef) {} FormalPowerSeries(const std::initializer_list<T> init) : coef(init.begin(), init.end()) {} template <typename InputIter> explicit FormalPowerSeries(const InputIter first, const InputIter last) : coef(first, last) {} inline const T& operator[](const int term) const { return coef[term]; } inline T& operator[](const int term) { return coef[term]; } using Mult = std::function<std::vector<T>(const std::vector<T>&, const std::vector<T>&)>; using Sqrt = std::function<bool(const T&, T*)>; static void set_mult(const Mult mult) { get_mult() = mult; } static void set_sqrt(const Sqrt sqrt) { get_sqrt() = sqrt; } void resize(const int deg) { coef.resize(deg + 1, 0); } void shrink() { while (coef.size() > 1 && coef.back() == 0) coef.pop_back(); } int degree() const { return std::ssize(coef) - 1; } FormalPowerSeries& operator=(const std::vector<T>& coef_) { coef = coef_; return *this; } FormalPowerSeries& operator=(const FormalPowerSeries& x) = default; FormalPowerSeries& operator+=(const FormalPowerSeries& x) { const int deg_x = x.degree(); if (deg_x > degree()) resize(deg_x); for (int i = 0; i <= deg_x; ++i) { coef[i] += x[i]; } return *this; } FormalPowerSeries& operator-=(const FormalPowerSeries& x) { const int deg_x = x.degree(); if (deg_x > degree()) resize(deg_x); for (int i = 0; i <= deg_x; ++i) { coef[i] -= x[i]; } return *this; } FormalPowerSeries& operator*=(const T x) { for (T& e : coef) e *= x; return *this; } FormalPowerSeries& operator*=(const FormalPowerSeries& x) { return *this = get_mult()(coef, x.coef); } FormalPowerSeries& operator/=(const T x) { assert(x != 0); return *this *= static_cast<T>(1) / x; } FormalPowerSeries& operator/=(const FormalPowerSeries& x) { const int n = degree() - x.degree() + 1; if (n <= 0) return *this = FormalPowerSeries(); const std::vector<T> tmp = get_mult()( std::vector<T>(coef.rbegin(), std::next(coef.rbegin(), n)), FormalPowerSeries( x.coef.rbegin(), std::next(x.coef.rbegin(), std::min(x.degree() + 1, n))) .inv(n - 1).coef); return *this = FormalPowerSeries(std::prev(tmp.rend(), n), tmp.rend()); } FormalPowerSeries& operator%=(const FormalPowerSeries& x) { if (x.degree() == 0) return *this = FormalPowerSeries{0}; *this -= *this / x * x; resize(x.degree() - 1); return *this; } FormalPowerSeries& operator<<=(const int n) { coef.insert(coef.begin(), n, 0); return *this; } FormalPowerSeries& operator>>=(const int n) { if (degree() < n) return *this = FormalPowerSeries(); coef.erase(coef.begin(), coef.begin() + n); return *this; } bool operator==(FormalPowerSeries x) const { x.shrink(); FormalPowerSeries y = *this; y.shrink(); return x.coef == y.coef; } FormalPowerSeries operator+() const { return *this; } FormalPowerSeries operator-() const { FormalPowerSeries res = *this; for (T& e : res.coef) e = -e; return res; } FormalPowerSeries operator+(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) += x; } FormalPowerSeries operator-(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) -= x; } FormalPowerSeries operator*(const T x) const { return FormalPowerSeries(*this) *= x; } FormalPowerSeries operator*(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) *= x; } FormalPowerSeries operator/(const T x) const { return FormalPowerSeries(*this) /= x; } FormalPowerSeries operator/(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) /= x; } FormalPowerSeries operator%(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) %= x; } FormalPowerSeries operator<<(const int n) const { return FormalPowerSeries(*this) <<= n; } FormalPowerSeries operator>>(const int n) const { return FormalPowerSeries(*this) >>= n; } T horner(const T x) const { return std::accumulate( coef.rbegin(), coef.rend(), static_cast<T>(0), [x](const T l, const T r) -> T { return l * x + r; }); } FormalPowerSeries differential() const { const int deg = degree(); assert(deg >= 0); FormalPowerSeries res(std::max(deg - 1, 0)); for (int i = 1; i <= deg; ++i) { res[i - 1] = coef[i] * i; } return res; } FormalPowerSeries exp(const int deg) const { assert(coef[0] == 0); const int n = coef.size(); const FormalPowerSeries one{1}; FormalPowerSeries res = one; for (int i = 1; i <= deg; i <<= 1) { res *= FormalPowerSeries(coef.begin(), std::next(coef.begin(), std::min(n, i << 1))) - res.log((i << 1) - 1) + one; res.coef.resize(i << 1); } res.resize(deg); return res; } FormalPowerSeries exp() const { return exp(degree()); } FormalPowerSeries inv(const int deg) const { assert(coef[0] != 0); const int n = coef.size(); FormalPowerSeries res{static_cast<T>(1) / coef[0]}; for (int i = 1; i <= deg; i <<= 1) { res = res + res - res * res * FormalPowerSeries( coef.begin(), std::next(coef.begin(), std::min(n, i << 1))); res.coef.resize(i << 1); } res.resize(deg); return res; } FormalPowerSeries inv() const { return inv(degree()); } FormalPowerSeries log(const int deg) const { assert(coef[0] == 1); FormalPowerSeries integrand = differential() * inv(deg - 1); integrand.resize(deg); for (int i = deg; i > 0; --i) { integrand[i] = integrand[i - 1] / i; } integrand[0] = 0; return integrand; } FormalPowerSeries log() const { return log(degree()); } FormalPowerSeries pow(long long exponent, const int deg) const { const int n = coef.size(); if (exponent == 0) { FormalPowerSeries res(deg); if (deg != -1) [[unlikely]] res[0] = 1; return res; } assert(deg >= 0); for (int i = 0; i < n; ++i) { if (coef[i] == 0) continue; if (i > deg / exponent) break; const long long shift = exponent * i; T tmp = 1, base = coef[i]; for (long long e = exponent; e > 0; e >>= 1) { if (e & 1) tmp *= base; base *= base; } const FormalPowerSeries res = ((*this >> i) / coef[i]).log(deg - shift); return ((res * exponent).exp(deg - shift) * tmp) << shift; } return FormalPowerSeries(deg); } FormalPowerSeries pow(const long long exponent) const { return pow(exponent, degree()); } FormalPowerSeries mod_pow(long long exponent, const FormalPowerSeries& md) const { const int deg = md.degree() - 1; if (deg < 0) [[unlikely]] return FormalPowerSeries(-1); const FormalPowerSeries inv_rev_md = FormalPowerSeries(md.coef.rbegin(), md.coef.rend()).inv(); const auto mod_mult = [&md, &inv_rev_md, deg]( FormalPowerSeries* multiplicand, const FormalPowerSeries& multiplier) -> void { *multiplicand *= multiplier; if (deg < multiplicand->degree()) { const int n = multiplicand->degree() - deg; const FormalPowerSeries quotient = FormalPowerSeries(multiplicand->coef.rbegin(), std::next(multiplicand->coef.rbegin(), n)) * FormalPowerSeries( inv_rev_md.coef.begin(), std::next(inv_rev_md.coef.begin(), std::min(deg + 2, n))); *multiplicand -= FormalPowerSeries(std::prev(quotient.coef.rend(), n), quotient.coef.rend()) * md; multiplicand->resize(deg); } multiplicand->shrink(); }; FormalPowerSeries res{1}, base = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) mod_mult(&res, base); mod_mult(&base, base); } return res; } FormalPowerSeries sqrt(const int deg) const { const int n = coef.size(); if (coef[0] == 0) { for (int i = 1; i < n; ++i) { if (coef[i] == 0) continue; if (i & 1) return FormalPowerSeries(-1); const int shift = i >> 1; if (deg < shift) break; FormalPowerSeries res = (*this >> i).sqrt(deg - shift); if (res.coef.empty()) return FormalPowerSeries(-1); res <<= shift; res.resize(deg); return res; } return FormalPowerSeries(deg); } T s; if (!get_sqrt()(coef.front(), &s)) return FormalPowerSeries(-1); FormalPowerSeries res{s}; const T half = static_cast<T>(1) / 2; for (int i = 1; i <= deg; i <<= 1) { res = (FormalPowerSeries(coef.begin(), std::next(coef.begin(), std::min(n, i << 1))) * res.inv((i << 1) - 1) + res) * half; } res.resize(deg); return res; } FormalPowerSeries sqrt() const { return sqrt(degree()); } FormalPowerSeries translate(const T c) const { const int n = coef.size(); std::vector<T> fact(n, 1), inv_fact(n, 1); for (int i = 1; i < n; ++i) { fact[i] = fact[i - 1] * i; } inv_fact[n - 1] = static_cast<T>(1) / fact[n - 1]; for (int i = n - 1; i > 0; --i) { inv_fact[i - 1] = inv_fact[i] * i; } std::vector<T> g(n), ex(n); for (int i = 0; i < n; ++i) { g[i] = coef[i] * fact[i]; } std::reverse(g.begin(), g.end()); T pow_c = 1; for (int i = 0; i < n; ++i) { ex[i] = pow_c * inv_fact[i]; pow_c *= c; } const std::vector<T> conv = get_mult()(g, ex); FormalPowerSeries res(n - 1); for (int i = 0; i < n; ++i) { res[i] = conv[n - 1 - i] * inv_fact[i]; } return res; } private: static Mult& get_mult() { static Mult mult = [](const std::vector<T>& a, const std::vector<T>& b) -> std::vector<T> { const int n = a.size(), m = b.size(); std::vector<T> res(n + m - 1, 0); for (int i = 0; i < n; ++i) { for (int j = 0; j < m; ++j) { res[i + j] += a[i] * b[j]; } } return res; }; return mult; } static Sqrt& get_sqrt() { static Sqrt sqrt = [](const T&, T*) -> bool { return false; }; return sqrt; } }; } // namespace emthrm #line 7 "include/emthrm/math/formal_power_series/bernoulli_number.hpp" namespace emthrm { template <typename T> std::vector<T> bernoulli_number(const int n) { FormalPowerSeries<T> bernoulli(n); bernoulli[0] = 1; for (int i = 1; i <= n; ++i) { bernoulli[i] = bernoulli[i - 1] / (i + 1); } bernoulli = bernoulli.inv(n); T fact = 1; for (int i = 0; i <= n; ++i) { bernoulli[i] *= fact; fact *= i + 1; } return bernoulli.coef; } } // namespace emthrm #line 9 "include/emthrm/math/formal_power_series/faulhaber_by_fps.hpp" namespace emthrm { template <unsigned int T> MInt<T> faulhaber_by_fps(const long long n, const int k) { using ModInt = MInt<T>; if (n <= 1) [[unlikely]] return 0; if (k == 0) [[unlikely]] return n - 1; ModInt::init(k + 1); const std::vector<ModInt> bernoulli = bernoulli_number<ModInt>(k); ModInt res = 0, p = 1; for (int i = k; i >= 0; --i) { p *= n; res += ModInt::nCk(k + 1, i) * bernoulli[i] * p; } return res / (k + 1); } } // namespace emthrm #line 14 "test/math/formal_power_series/faulhaber_by_fps.test.cpp" int main() { constexpr int MOD = 1000000007; using ModInt = emthrm::MInt<MOD>; emthrm::FormalPowerSeries<ModInt>::set_mult( [](const std::vector<ModInt>& a, const std::vector<ModInt>& b) -> std::vector<ModInt> { return emthrm::mod_convolution(a, b); }); long long n; int k; std::cin >> n >> k; std::cout << emthrm::faulhaber_by_fps<MOD>(n + 1, k) << '\n'; return 0; }