#pragma once #ifndef INCLUDED_MODINT_HPP #define INCLUDED_MODINT_HPP #include template class modint { using u32 = std::uint_fast32_t; using u64 = std::uint_fast64_t; using i64 = std::int_fast64_t; inline u64 apply(i64 x) { return (x < 0 ? x + Modulus : x); }; public: u64 a; static constexpr u64 mod = Modulus; constexpr modint(const i64& x = 0) noexcept: a(apply(x % (i64)Modulus)) {} constexpr modint operator+(const modint& rhs) const noexcept { return modint(*this) += rhs; } constexpr modint operator-(const modint& rhs) const noexcept { return modint(*this) -= rhs; } constexpr modint operator*(const modint& rhs) const noexcept { return modint(*this) *= rhs; } constexpr modint operator/(const modint& rhs) const noexcept { return modint(*this) /= rhs; } constexpr modint operator^(const u64& k) const noexcept { return modint(*this) ^= k; } constexpr modint operator^(const modint& k) const noexcept { return modint(*this) ^= k.value(); } constexpr modint operator-() const noexcept { return modint(Modulus - a); } constexpr modint operator++() noexcept { return (*this) = modint(*this) + 1; } constexpr modint operator--() noexcept { return (*this) = modint(*this) - 1; } const bool operator==(const modint& rhs) const noexcept { return a == rhs.a; }; const bool operator!=(const modint& rhs) const noexcept { return a != rhs.a; }; const bool operator<=(const modint& rhs) const noexcept { return a <= rhs.a; }; const bool operator>=(const modint& rhs) const noexcept { return a >= rhs.a; }; const bool operator<(const modint& rhs) const noexcept { return a < rhs.a; }; const bool operator>(const modint& rhs) const noexcept { return a > rhs.a; }; constexpr modint& operator+=(const modint& rhs) noexcept { a += rhs.a; if (a >= Modulus) a -= Modulus; return *this; } constexpr modint& operator-=(const modint& rhs) noexcept { if (a < rhs.a) a += Modulus; a -= rhs.a; return *this; } constexpr modint& operator*=(const modint& rhs) noexcept { a = a * rhs.a % Modulus; return *this; } constexpr modint& operator/=(modint rhs) noexcept { u64 exp = Modulus - 2; while (exp) { if (exp % 2) (*this) *= rhs; rhs *= rhs; exp /= 2; } return *this; } constexpr modint& operator^=(u64 k) noexcept { auto b = modint(1); while(k) { if(k & 1) b = b * (*this); (*this) *= (*this); k >>= 1; } return (*this) = b; } constexpr modint& operator=(const modint& rhs) noexcept { a = rhs.a; return (*this); } constexpr u64& value() noexcept { return a; } constexpr const u64& value() const noexcept { return a; } explicit operator bool() const { return a; } explicit operator u32() const { return a; } const modint inverse() const { return modint(1) / *this; } const modint pow(i64 k) const { return modint(*this) ^ k; } friend std::ostream& operator<<(std::ostream& os, const modint& p) { return os << p.a; } friend std::istream& operator>>(std::istream& is, modint& p) { u64 t; is >> t; p = modint(t); return is; } }; #endif #pragma once #ifndef INCLUDED_POLYNOMIAL_HPP #define INCLUDED_POLYNOMIAL_HPP #include #include template class polynomial: public std::vector { public: using std::vector::vector; using value_type = typename std::vector::value_type; using reference = typename std::vector::reference; using const_reference = typename std::vector::const_reference; using size_type = typename std::vector::size_type; private: T eval(T x) const { T ret = (*this)[0], tmp = x; for(int i = 1; i < this->size(); i++) { ret = ret + (*this)[i] * tmp; tmp = tmp * x; } return ret; } public: polynomial(): std::vector(1, T{}) {} polynomial(const std::vector& p): std::vector(p) {} polynomial operator+(const polynomial& r) const { return polynomial(*this) += r; } polynomial operator-(const polynomial& r) const { return polynomial(*this) -= r; } polynomial operator*(const_reference r) const { return polynomial(*this) *= r; } polynomial operator/(const_reference r) const { return polynomial(*this) /= r; } polynomial operator<<(size_type r) const { return polynomial(*this) <<= r; } polynomial operator>>(size_type r) const { return polynomial(*this) >>= r; } polynomial operator-() const { polynomial ret(this->size()); for(int i = 0; i < this->size(); i++) ret[i] = -(*this)[i]; return ret; } polynomial& operator+=(const polynomial& r) { if(r.size() > this->size()) this->resize(r.size()); for(int i = 0; i < r.size(); i++) (*this)[i] = (*this)[i] + r[i]; return *this; } polynomial& operator-=(const polynomial& r) { if(r.size() > this->size()) this->resize(r.size()); for(int i = 0; i < r.size(); i++) (*this)[i] = (*this)[i] - r[i]; return *this; } polynomial& operator*=(const_reference r) { for(int i = 0; i < this->size(); i++) (*this)[i] = (*this)[i] * r; return *this; } polynomial& operator/=(const_reference r) { for(int i = 0; i < this->size(); i++) (*this)[i] = (*this)[i] / r; return *this; } polynomial& operator<<=(size_type r) { this->insert(begin(*this), r, T{}); return *this; } polynomial& operator>>=(size_type r) { if(r >= this->size()) clear(); else this->erase(begin(*this), begin(*this) + r); return *this; } polynomial differential(size_type k) const { polynomial ret(*this); for(int i = 0; i < k; i++) ret = ret.differential(); return ret; } polynomial differential() const { if(degree() < 1) return polynomial(); polynomial ret(this->size() - 1); for(int i = 1; i < this->size(); i++) ret[i - 1] = (*this)[i] * T{i}; return ret; } polynomial integral(size_type k) const { polynomial ret(*this); for(int i = 0; i < k; i++) ret = ret.integral(); return ret; } polynomial integral() const { polynomial ret(this->size() + 1); for(int i = 0; i < this->size(); i++) ret[i + 1] = (*this)[i] / T{i + 1}; return ret; } polynomial prefix(size_type size) const { if(size == 0) return polynomial(); return polynomial(begin(*this), begin(*this) + std::min(this->size(), size)); } void shrink() { while(this->size() > 1 and this->back() == T{}) this->pop_back(); } T operator()(T x) const { return eval(x); } size_type degree() const { return this->size() - 1; } void clear() { this->assign(1, T{}); } }; #endif #pragma once #ifndef INCLUDED_NUMBER_THEORITIC_TRANSFORM_HPP #define INCLUDED_NUMBER_THEORITIC_TRANSFORM_HPP template class number_theoritic_transform: public polynomial { public: using polynomial::polynomial; using value_type = typename polynomial::value_type; using reference = typename polynomial::reference; using const_reference = typename polynomial::const_reference; using size_type = typename polynomial::size_type; private: void ntt(number_theoritic_transform& a) const { int N = a.size(); static std::vector dw; if(dw.size() < N) { int n = dw.size(); dw.resize(N); for(int i = n; i < N; i++) dw[i] = -(T(primitive_root) ^ ((T::mod - 1) >> i + 2)); } for(int m = N; m >>= 1;) { T w = 1; for(int s = 0, k = 0; s < N; s += 2 * m) { for(int i = s, j = s + m; i < s + m; i++, j++) { T x = a[i], y = a[j] * w; a[i] = x + y; a[j] = x - y; } w *= dw[__builtin_ctz(++k)]; } } } void intt(number_theoritic_transform& a) const { int N = a.size(); static std::vector idw; if(idw.size() < N) { int n = idw.size(); idw.resize(N); for(int i = n; i < N; i++) idw[i] = (-(T(primitive_root) ^ ((T::mod - 1) >> i + 2))).inverse(); } for(int m = 1; m < N; m *= 2) { T w = 1; for(int s = 0, k = 0; s < N; s += 2 * m) { for(int i = s, j = s + m; i < s + m; i++, j++) { T x = a[i], y = a[j]; a[i] = x + y; a[j] = (x - y) * w; } w *= idw[__builtin_ctz(++k)]; } } } void transform(number_theoritic_transform& a, bool inverse = false) const { size_type n = 0; while((1 << n) < a.size()) n++; size_type N = 1 << n; a.resize(N); if(!inverse) { ntt(a); } else { intt(a); T inv = T(N).inverse(); for(int i = 0; i < a.size(); i++) a[i] *= inv; } } number_theoritic_transform convolution(const number_theoritic_transform& ar, const number_theoritic_transform& br) const { size_type size = ar.degree() + br.degree() + 1; number_theoritic_transform a(ar), b(br); a.resize(size); b.resize(size); transform(a, false); transform(b, false); for(int i = 0; i < a.size(); i++) a[i] *= b[i]; transform(a, true); a.resize(size); return a; } public: number_theoritic_transform(const polynomial& p): polynomial(p) {} number_theoritic_transform operator*(const_reference r) const { return number_theoritic_transform(*this) *= r; } number_theoritic_transform& operator*=(const_reference r) { for(int i = 0; i < this->size(); i++) (*this)[i] = (*this)[i] * r; return *this; } number_theoritic_transform operator*(const number_theoritic_transform& r) const { return number_theoritic_transform(*this) *= r; } number_theoritic_transform& operator*=(const number_theoritic_transform& r) { return (*this) = convolution((*this), r); } }; #endif #ifndef INCLUDED_ARBITARY_MOD_NUMBER_THEORITIC_TRANSFORM_HPP #define INCLUDED_ARBITARY_MOD_NUMBER_THEORITIC_TRANSFORM_HPP namespace amnttlib { using u64 = std::uint_fast64_t; // https://lumakernel.github.io/ecasdqina/math/FFT/NTT constexpr u64 ntt_primes[][2] = { {1224736769, 3}, // 2^24 * 73 + 1, {1053818881, 7}, // 2^20 * 3 * 5 * 67 + 1 {1051721729, 6}, // 2^20 * 17 * 59 + 1 {1045430273, 3}, // 2^20 * 997 + 1 {1012924417, 5}, // 2^21 * 3 * 7 * 23 + 1 {1007681537, 3}, // 2^20 * 31^2 + 1 {1004535809, 3}, // 2^21 * 479 + 1 {998244353, 3}, // 2^23 * 7 * 17 + 1 {985661441, 3}, // 2^22 * 5 * 47 + 1 {976224257, 3}, // 2^20 * 7^2 * 19 + 1 {975175681, 17}, // 2^21 * 3 * 5 * 31 + 1 {962592769, 7}, // 2^21 * 3^3 * 17 + 1 {950009857, 7}, // 2^21 * 4 * 151 + 1 {943718401, 7}, // 2^22 * 3^2 * 5^2 + 1 {935329793, 3}, // 2^22 * 223 + 1 {924844033, 5}, // 2^21 * 3^2 * 7^2 + 1 {469762049, 3}, // 2^26 * 7 + 1 {167772161, 3}, // 2^25 * 5 + 1 }; }; template class arbitary_mod_number_theoritic_transform: public polynomial { public: using polynomial::polynomial; using value_type = typename polynomial::value_type; using reference = typename polynomial::reference; using const_reference = typename polynomial::const_reference; using size_type = typename polynomial::size_type; using amntt = arbitary_mod_number_theoritic_transform; using m1 = modint; using m2 = modint; using m3 = modint; private: amntt convolution(const amntt& ar, const amntt& br) { number_theoritic_transform ntt1_a(ar.size()), ntt1_b(br.size()); number_theoritic_transform ntt2_a(ar.size()), ntt2_b(br.size()); number_theoritic_transform ntt3_a(ar.size()), ntt3_b(br.size()); for(int i = 0; i < ar.size(); i++) { ntt1_a[i] = m1(ar[i].value()); ntt2_a[i] = m2(ar[i].value()); ntt3_a[i] = m3(ar[i].value()); } for(int i = 0; i < br.size(); i++) { ntt1_b[i] = m1(br[i].value()); ntt2_b[i] = m2(br[i].value()); ntt3_b[i] = m3(br[i].value()); } auto x = ntt1_a * ntt1_b; auto y = ntt2_a * ntt2_b; auto z = ntt3_a * ntt3_b; amntt ret(x.size()); const m2 m1_inv_m2 = m2(MOD_1).inverse(); const m3 m12_inv_m3 = (m3(MOD_1) * m3(MOD_2)).inverse(); const T m12 = T(MOD_1) * T(MOD_2); for(int i = 0; i < ret.size(); i++) { m2 v1 = (m2(y[i].value()) - m2(x[i].value())) * m1_inv_m2; m3 v2 = (m3(z[i].value()) - (m3(x[i].value()) + m3(MOD_1) * m3(v1.value()))) * m12_inv_m3; ret[i] = (T(x[i].value()) + T(MOD_1) * T(v1.value()) + m12 * T(v2.value())); } ret.resize(ar.degree() + br.degree() + 1); return ret; } public: arbitary_mod_number_theoritic_transform(const polynomial& p): polynomial(p) {} amntt operator*(const_reference r) const { return amntt(*this) *= r; } amntt& operator*=(const_reference r) { for(int i = 0; i < this->size(); i++) (*this)[i] = (*this)[i] * r; return *this; } amntt operator*(const amntt& r) const { return amntt(*this) *= r; } amntt& operator*=(const amntt& r) { return (*this) = convolution((*this), r); } }; #endif #pragma once #ifndef INCLUDED_FORMAL_POWER_SERIES_HPP #define INCLUDED_FORMAL_POWER_SERIES_HPP #include #include template class formal_power_series: public T { public: using T::T; using value_type = typename T::value_type; using reference = typename T::reference; using const_reference = typename T::const_reference; using size_type = typename T::size_type; private: formal_power_series(): T(1) {} formal_power_series(const T& p): T(p) {} public: formal_power_series inverse() const { assert((*this)[0] != value_type{}); formal_power_series ret(1, (*this)[0].inverse()); for(int i = 1; i < this->size(); i <<= 1) { auto tmp = ret * this->prefix(i << 1); for(int j = 0; j < i; j++) { tmp[j] = value_type{}; if(j + i < tmp.size()) tmp[j + i] *= value_type(-1); } tmp = tmp * ret; for(int j = 0; j < i; j++) tmp[j] = ret[j]; ret = std::move(tmp).prefix(i << 1); } return ret.prefix(this->size()); } formal_power_series log() const { assert((*this)[0] == value_type(1)); return (formal_power_series(this->differential()) * this->inverse()).integral().prefix(this->size()); } formal_power_series exp() const { assert((*this)[0] == value_type{}); formal_power_series f(1, value_type(1)), g(1, value_type(1)); for(int i = 1; i < this->size(); i <<= 1) { g = (g * value_type(2) - f * g * g).prefix(i); formal_power_series q = this->differential().prefix(i - 1); formal_power_series w = (q + g * (f.differential() - f * q)).prefix((i << 1) - 1); f = (f + f * (*this - w.integral()).prefix(i << 1)).prefix(i << 1); } return f.prefix(this->size()); } formal_power_series pow(size_type k) const { for(size_type i = 0; i < this->size(); i++) { if((*this)[i] != value_type{}) { value_type inv = (*this)[i].inverse(); formal_power_series f(*this * inv); formal_power_series g(f >> i); g = formal_power_series(g.log() * value_type(k)).exp() * (*this)[i].pow(k); if(i * k > this->size()) return formal_power_series(this->size()); return (g << (i * k)).prefix(this->size()); } } return *this; } }; #endif using fps = formal_power_series>>; int main() { int n, d, k; scanf("%d%d%d", &n, &d, &k); fps p(k + 1); for(int i = 1; i <= std::min(d, k); i++) p[i] = 1; printf("%d\n", p.pow(n)[k]); return 0; }