#include #include #include #include #include #include #include #include #include using namespace std; #define reE(i,a,b) for(auto (i)=(a);(i)<=(b);(i)++) #define rE(i,b) reE(i,0,b) #define reT(i,a,b) for(auto (i)=(a);(i)<(b);(i)++) #define rT(i,b) reT(i,0,b) #define rep(i,a,b) reE(i,a,b); #define rev(i,a,b) for(auto (i)=(b)-1;(i)>=(a);(i)--) #define itr(i,b) for(auto (i)=(b).begin();(i)!=(b).end();++(i)) #define LL long long #define BUFFSIZE 1300LL #define MAX_LOGN 60 typedef long long ll; typedef pair Pii; #define FOR(i,n) for(int i = 0; i < (n); i++) #define sz(c) ((int)(c).size()) #define ten(x) ((int)1e##x) template T extgcd(T a, T b, T& x, T& y) { for (T u = y = 1, v = x = 0; a;) { T q = b / a; swap(x -= q * u, u); swap(y -= q * v, v); swap(b -= q * a, a); } return b; } template T mod_inv(T a, T m) { T x, y; extgcd(a, m, x, y); return (m + x % m) % m; } ll mod_pow(ll a, ll n, ll mod) { ll ret = 1; ll p = a % mod; while (n) { if (n & 1) ret = ret * p % mod; p = p * p % mod; n >>= 1; } return ret; } template class NTT { public: int get_mod() const { return mod; } void _ntt(vector& a, int sign) { const int n = sz(a); assert((n ^ (n&-n)) == 0); //n = 2^k const int g = 3; //g is primitive root of mod int h = (int)mod_pow(g, (mod - 1) / n, mod); // h^n = 1 if (sign == -1) h = (int)mod_inv(h, mod); //h = h^-1 % mod //bit reverse 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) { const int m2 = 2 * m; const ll base = mod_pow(h, n / m2, mod); ll w = 1; FOR(x, m) { for (int s = x; s < n; s += m2) { ll u = a[s]; ll d = a[s + m] * w; if (d>mod)d %= mod; a[s] = u + d; if (a[s] >= mod) a[s] -= mod; a[s + m] = u - d; if (a[s + m] < 0) a[s + m] += mod; } w = w * base; if (w>mod)w %= mod; } } for (auto& x : a) if (x < 0) x += mod; } void ntt(vector& input) { _ntt(input, 1); } void intt(vector& input) { _ntt(input, -1); const int n_inv = mod_inv(sz(input), mod); for (auto& x : input){ x = x * n_inv; if (x > mod)x %= mod; } } // 畳み込み演算を行う vector convolution(const vector& a, const vector& b){ int ntt_size = 1; while (ntt_size < sz(a) + sz(b)) ntt_size *= 2; vector _a = a, _b = b; _a.resize(ntt_size); _b.resize(ntt_size); ntt(_a); ntt(_b); FOR(i, ntt_size){ _a[i] *= _b[i]; if (_a[i] > mod)_a[i] %= mod; } intt(_a); return _a; } }; ll garner(vector mr, int mod){ mr.emplace_back(mod, 0); vector coffs(sz(mr), 1); vector constants(sz(mr), 0); FOR(i, sz(mr) - 1){ // coffs[i] * v + constants[i] == mr[i].second (mod mr[i].first) を解く ll v = (mr[i].second - constants[i]) * mod_inv(coffs[i], mr[i].first) % mr[i].first; if (v < 0) v += mr[i].first; for (int j = i + 1; j < sz(mr); j++) { (constants[j] += coffs[j] * v) %= mr[j].first; (coffs[j] *= mr[i].first) %= mr[j].first; } } return constants[sz(mr) - 1]; } typedef NTT<167772161, 3> NTT_1; typedef NTT<469762049, 3> NTT_2; typedef NTT<1224736769, 3> NTT_3; //任意のmodで畳み込み演算 O(n log n) vector int32mod_convolution(vector a, vector b, int mod){ for (auto& x : a) x %= mod; for (auto& x : b) x %= mod; NTT_1 ntt1; NTT_2 ntt2; NTT_3 ntt3; auto x = ntt1.convolution(a, b); auto y = ntt2.convolution(a, b); auto z = ntt3.convolution(a, b); vector ret(sz(x)); vector mr(3); FOR(i, sz(x)){ mr[0].first = ntt1.get_mod(), mr[0].second = (int)x[i]; mr[1].first = ntt2.get_mod(), mr[1].second = (int)y[i]; mr[2].first = ntt3.get_mod(), mr[2].second = (int)z[i]; ret[i] = garner(mr, mod); } return ret; } // garnerのアルゴリズムを直書きしたversion,速い vector fast_int32mod_convolution(vector a, vector b, int mod){ for (auto& x : a) x %= mod; for (auto& x : b) x %= mod; NTT_1 ntt1; NTT_2 ntt2; NTT_3 ntt3; assert(ntt1.get_mod() < ntt2.get_mod() && ntt2.get_mod() < ntt3.get_mod()); auto x = ntt1.convolution(a, b); auto y = ntt2.convolution(a, b); auto z = ntt3.convolution(a, b); // garnerのアルゴリズムを極力高速化した const ll m1 = ntt1.get_mod(), m2 = ntt2.get_mod(), m3 = ntt3.get_mod(); const ll m1_inv_m2 = mod_inv(m1, m2); const ll m12_inv_m3 = mod_inv(m1 * m2, m3); const ll m12_mod = m1 * m2 % mod; vector ret(sz(x)); FOR(i, sz(x)){ ll v1 = (y[i] - x[i]) * m1_inv_m2 % m2; if (v1 < 0) v1 += m2; ll v2 = (z[i] - (x[i] + m1 * v1) % m3) * m12_inv_m3 % m3; if (v2 < 0) v2 += m3; ll constants3 = (x[i] + m1 * v1 + m12_mod * v2) % mod; if (constants3 < 0) constants3 += mod; ret[i] = constants3; } return ret; } #define MOD 1000000007 template class Mr{ public: vector first; vector C; vector> bin; vector t; vector lw, rw;//M+1個の作業用 vector lu, ru;//M個の作業用 T zero, one; int M; //n(1,,,2M)をn(1,,,M)に修正、O(M^2) void form(vector &n){ for (int i = 0; i < M; i++){ lu[i] = t[i].val; ru[i] = n[2 * M - i].val; } auto b = fast_int32mod_convolution(lu, ru, MOD); for (int i = 0; i < M; i++){ rw[i] = b[i]; lw[i + 1] = C[M - i - 1].val; } rw[M] = 0; lw[0] = 0; auto bb = fast_int32mod_convolution(lw, rw, MOD); n[0] = zero; for (int i = 1; i <= M; i++){ n[i] = (n[i].val + bb[2 * M - i]) % MOD; n[M + i] = zero; } } //lとrを足し合わせる、O(M^2) inline void add(vector &l, vector &r, vector &ans){ lw[0] = 0; rw[0] = 0; for (int i = 1; i <= M; i++){ lw[i] = l[i].val; rw[i] = r[i].val; } auto tmp = fast_int32mod_convolution(lw, rw, MOD); reE(i, 1, 2 * M)ans[i].val = tmp[i]; form(ans); } //初期化、O(M*MAX_LOGN) Mr(const vector& f, const vector& c, int m, T e1, T e0){ M = m; first.reserve(M + 1); C.reserve(M); lw.resize(M + 1); rw.resize(M + 1); lu.resize(M); ru.resize(M); zero = e0, one = e1; first.push_back(zero); rT(i, M){ first.push_back(f[i]); C.push_back(c[i]); } t.resize(M); t[0] = one; for (int i = 0; i < M; i++) for (int j = i + 1; j < M; j++){ t[j] = t[j] + C[M - (j - i)] * t[i]; } bin.resize(MAX_LOGN); rT(i, MAX_LOGN)bin[i].resize(2 * M + 1); rE(i, 2 * M)bin[0][i] = zero; bin[0][1] = one; reT(i, 1, MAX_LOGN){ add(bin[i - 1], bin[i - 1], bin[i]); } } //N項目の計算、戻り値がTの形であることに注意、O(M^2*logN) vector calc(LL n){ n--; vector tmp, result = bin[0]; for (int b = 0; n; b++, n >>= 1) if (1 & n){ tmp = result; add(tmp, bin[b], result); } //reE(i, 1, M)ans = ans + (result[i] * first[i]); return result; } }; //テンプレート、デフォルトコンストラクタのオーバーロードを忘れない struct X{ LL val; X(LL v){ val = v; } X(){ val = 0; } LL operator=(const X &another){ return val = another.val; } LL operator*(const X &another)const{ return (val*another.val) % MOD; } LL operator+(const X &another)const{ return (val + another.val) % MOD; } }; #define DPBUFF 700 LL comb[BUFFSIZE + 1]; LL prevDP[DPBUFF + 1][6]; LL nextDP[DPBUFF + 1][6]; LL dp[2 * BUFFSIZE + 1] = { 1 }; LL sumc[BUFFSIZE + 1] = { 0 }; vector sosuu = { 2, 3, 5, 7, 11, 13 }, gousei = { 4, 6, 8, 9, 10, 12 }; int p, c; vector dpcalc(vectors, int pc){ for (int i = 0; ires(DPBUFF, 0); for (int i = 0; i < DPBUFF; i++){ for (int j = 0; j < 6; j++) res[i] = (res[i] + prevDP[i][j]); res[i] %= MOD; } return res; } vector C(BUFFSIZE + 1); int main(void){ LL N; cin >> N >> p >> c; //combの計算 auto xx = dpcalc(sosuu, p); auto yy = dpcalc(gousei, c); auto zz = fast_int32mod_convolution(xx, yy, MOD); for (int i = 0; iMOD)dp[i] %= MOD; } } //漸化式の係数部分 for (int i = 1; i <= BUFFSIZE; i++){ C[BUFFSIZE - i] = X(comb[i]); } for (int i = BUFFSIZE - 1; i >= 0; i--){ sumc[i] = (sumc[i + 1] + comb[i]) % MOD; } LL res = 0; if (N < BUFFSIZE + 100){ for (LL i = 0; i < min(N, BUFFSIZE); i++){ res = (res + dp[i] * sumc[min(N, BUFFSIZE) - i]); if (res>MOD)res %= MOD; }; } else{ // cout << "M" << endl; Mr mr(C, C, BUFFSIZE, X(1), X(0)); auto cc = mr.calc(N + 1 - BUFFSIZE); // cout << "M" << endl; for (int i = 0; i < BUFFSIZE; i++) if (sumc[BUFFSIZE - i]){ LL x = 0; for (int j = 0; j < BUFFSIZE; j++){ x = (x + dp[i + j] * cc[j + 1].val); if (x>MOD)x %= MOD; } res = (res + x*sumc[BUFFSIZE - i]); if (res>MOD)res %= MOD; } } cout << res << endl; return(0); }