#define PROBLEM "https://yukicoder.me/problems/no/215" #include #include #include // CUT begin template struct ModInt { using lint = long long; static int get_mod() { return mod; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&](){ std::set fac; int v = mod - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < mod; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).power((mod - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val; constexpr ModInt() : val(0) {} constexpr ModInt &_setval(lint v) { val = (v >= mod ? v - mod : v); return *this; } constexpr ModInt(lint v) { _setval(v % mod + mod); } explicit operator bool() const { return val != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + mod); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); } constexpr ModInt operator-() const { return ModInt()._setval(mod - val); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % mod + x.val); } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); } constexpr bool operator==(const ModInt &x) const { return val == x.val; } constexpr bool operator!=(const ModInt &x) const { return val != x.val; } bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; is >> t; x = ModInt(t); return is; } friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { os << x.val; return os; } constexpr lint power(lint n) const { lint ans = 1, tmp = this->val; while (n) { if (n & 1) ans = ans * tmp % mod; tmp = tmp * tmp % mod; n /= 2; } return ans; } constexpr lint inv() const { return this->power(mod - 2); } constexpr ModInt operator^(lint n) const { return ModInt(this->power(n)); } constexpr ModInt &operator^=(lint n) { return *this = *this ^ n; } inline ModInt fac() const { static std::vector facs; int l0 = facs.size(); if (l0 > this->val) return facs[this->val]; facs.resize(this->val + 1); for (int i = l0; i <= this->val; i++) facs[i] = (i == 0 ? ModInt(1) : facs[i - 1] * ModInt(i)); return facs[this->val]; } ModInt doublefac() const { lint k = (this->val + 1) / 2; if (this->val & 1) return ModInt(k * 2).fac() / ModInt(2).power(k) / ModInt(k).fac(); else return ModInt(k).fac() * ModInt(2).power(k); } ModInt nCr(const ModInt &r) const { if (this->val < r.val) return ModInt(0); return this->fac() / ((*this - r).fac() * r.fac()); } ModInt sqrt() const { if (val == 0) return 0; if (mod == 2) return val; if (power((mod - 1) / 2) != 1) return 0; ModInt b = 1; while (b.power((mod - 1) / 2) == 1) b += 1; int e = 0, m = mod - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = power((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.power(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.power(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val, mod - x.val)); } }; #include #include #include #include using namespace std; // CUT begin // Integer convolution for arbitrary mod // with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class. // We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`. // input: a (size: n), b (size: m) // return: vector (size: n + m - 1) template vector nttconv(vector a, vector b, bool skip_garner = false); constexpr int nttprimes[3] = {998244353, 167772161, 469762049}; // Integer FFT (Fast Fourier Transform) for ModInt class // (Also known as Number Theoretic Transform, NTT) // is_inverse: inverse transform // ** Input size must be 2^n ** template void ntt(vector &a, bool is_inverse = false) { int n = a.size(); assert(__builtin_popcount(n) == 1); MODINT h = MODINT(MODINT::get_primitive_root()).power((MODINT::get_mod() - 1) / n); if (is_inverse) h = 1 / h; int i = 0; for (int j = 1; j < n - 1; j++) { for (int k = n >> 1; k > (i ^= k); k >>= 1); if (j < i) swap(a[i], a[j]); } for (int m = 1; m < n; m *= 2) { int m2 = 2 * m; MODINT base = h.power(n / m2), w = 1; for (int x = 0; x < m; x++) { for (int s = x; s < n; s += m2) { MODINT u = a[s], d = a[s + m] * w; a[s] = u + d, a[s + m] = u - d; } w *= base; } } if (is_inverse) { long long int n_inv = MODINT(n).inv(); for (auto &v : a) v *= n_inv; } } template vector> nttconv_(const vector &a, const vector &b) { int sz = a.size(); assert(a.size() == b.size() and __builtin_popcount(sz) == 1); vector> ap(sz), bp(sz); for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i]; if (a == b) { ntt(ap, false); bp = ap; } else { ntt(ap, false); ntt(bp, false); } for (int i = 0; i < sz; i++) ap[i] *= bp[i]; ntt(ap, true); return ap; } long long int extgcd_ntt_(long long int a, long long int b, long long int &x, long long int &y) { long long int d = a; if (b != 0) d = extgcd_ntt_(b, a % b, y, x), y -= (a / b) * x; else x = 1, y = 0; return d; } long long int modinv_ntt_(long long int a, long long int m) { long long int x, y; extgcd_ntt_(a, m, x, y); return (m + x % m) % m; } long long int garner_ntt_(int r0, int r1, int r2, int mod) { array rs = {r0, r1, r2, 0}; vector coffs(4, 1), constants(4, 0); for (int i = 0; i < 3; i++) { long long int v = (rs[i] - constants[i]) * modinv_ntt_(coffs[i], nttprimes[i]) % nttprimes[i]; if (v < 0) v += nttprimes[i]; for (int j = i + 1; j < 4; j++) { (constants[j] += coffs[j] * v) %= (j < 3 ? nttprimes[j] : mod); (coffs[j] *= nttprimes[i]) %= (j < 3 ? nttprimes[j] : mod); } } return constants.back(); } template vector nttconv(vector a, vector b, bool skip_garner) { int sz = 1, n = a.size(), m = b.size(); while (sz < n + m) sz <<= 1; if (sz <= 16) { vector ret(n + m - 1); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j]; } return ret; } int mod = MODINT::get_mod(); if (skip_garner or find(begin(nttprimes), end(nttprimes), mod) != end(nttprimes)) { a.resize(sz), b.resize(sz); if (a == b) { ntt(a, false); b = a; } else ntt(a, false), ntt(b, false); for (int i = 0; i < sz; i++) a[i] *= b[i]; ntt(a, true); a.resize(n + m - 1); } else { vector ai(sz), bi(sz); for (int i = 0; i < n; i++) ai[i] = a[i].val; for (int i = 0; i < m; i++) bi[i] = b[i].val; auto ntt0 = nttconv_(ai, bi); auto ntt1 = nttconv_(ai, bi); auto ntt2 = nttconv_(ai, bi); a.resize(n + m - 1); for (int i = 0; i < n + m - 1; i++) { a[i] = garner_ntt_(ntt0[i].val, ntt1[i].val, ntt2[i].val, mod); } } return a; } #include // Calculate [x^N](num(x) / den(x)) template Tp coefficient_of_rational_function(long long N, std::vector num, std::vector den) { assert(N >= 0); while (den.size() and den.back() == 0) den.pop_back(); assert(den.size()); int h = 0; while (den[h] == 0) h++; N += h; den.erase(den.begin(), den.begin() + h); if (den.size() == 1) { assert(N < int(num.size())); return num[N] / den[0]; } while (N) { std::vector g = den; for (size_t i = 1; i < g.size(); i += 2) { g[i] = -g[i]; } auto conv_num_g = nttconv(num, g); num.resize((conv_num_g.size() + 1 - (N & 1)) / 2); for (size_t i = 0; i < num.size(); i++) { num[i] = conv_num_g[i * 2 + (N & 1)]; } auto conv_den_g = nttconv(den, g); for (size_t i = 0; i < den.size(); i++) { den[i] = conv_den_g[i * 2]; } N >>= 1; } return num[0] / den[0]; } using mint = ModInt<1000000007>; #include using namespace std; vector gen_dp(vector v, int n) { vector> dp(n + 1, vector(v.back() * n + 1)); dp[0][0] = 1; for (auto x : v) { for (int i = n - 1; i >= 0; i--) { for (int j = 0; j < dp[i].size(); j++) if (dp[i][j]) { for (int k = 1; i + k <= n; k++) dp[i + k][j + x * k] += dp[i][j]; } } } return dp.back(); } int main() { long long N; int P, C; cin >> N >> P >> C; vector primes = gen_dp({2, 3, 5, 7, 11, 13}, P), composites = gen_dp({4, 6, 8, 9, 10, 12}, C); vector f = nttconv(primes, composites); vector denom = f; for (auto &x : denom) x = -x; denom[0] = 1; for (int i = f.size() - 1; i > 1; i--) f[i - 1] += f[i]; cout << coefficient_of_rational_function(N, f, denom) << '\n'; }