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#include "poly/power_projection.hpp"

#pragma once #include "poly/convolution.hpp" template <typename mint> vc<mint> power_projection_0_ntt(vc<mint> wt, vc<mint> f, int m) { assert(len(f) == len(wt) && f[0] == mint(0)); int n = 1; while (n < len(f)) n *= 2; for (auto& x: f) x = -x; f.resize(n), wt.resize(n); reverse(all(wt)); vc<mint>&P = wt, &Q = f; P.resize(4 * n), Q.resize(4 * n); vc<mint> W(n); { // bit reverse order vc<int> btr(n); int log = topbit(n); FOR(i, n) { btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (log - 1)); } int t = mint::ntt_info().fi; mint r = mint::ntt_info().se; mint dw = r.inverse().pow((1 << t) / (2 * n)); mint w = 1; for (auto& i: btr) { W[i] = w, w *= dw; } } int k = 1; while (n > 1) { /* FFT step 04.. -> 048c 15.. -> 159d .... -> 26ae .... -> 37bf */ auto doubling_y = [&](vc<mint>& A, int l, int r) -> void { mint z = W[k / 2].inverse(); vc<mint> f(k); FOR(i, l, r) { FOR(j, k) f[j] = A[2 * n * j + i]; ntt(f, 1); mint r = 1; FOR(j, 1, k) r *= z, f[j] *= r; ntt(f, 0); FOR(j, k) A[2 * n * (k + j) + i] = f[j]; } }; auto FFT_x = [&](vc<mint>& A, int l, int r) -> void { vc<mint> f(2 * n); FOR(j, l, r) { move(A.begin() + 2 * n * j, A.begin() + 2 * n * (j + 1), f.begin()); ntt(f, 0); move(all(f), A.begin() + 2 * n * j); } }; if (n <= k) { doubling_y(P, 0, n), doubling_y(Q, 1, n); FFT_x(P, 0, 2 * k), FFT_x(Q, 0, 2 * k); } else { FFT_x(P, 0, k), FFT_x(Q, 0, k); doubling_y(P, 0, 2 * n), doubling_y(Q, 0, 2 * n); } FOR(i, 2 * n * k) Q[i] += 1; FOR(i, 2 * n * k, 4 * n * k) Q[i] -= 1; /* 048c -> 0248???? 159d -> ....???? 26ae 37bf */ vc<mint> F(2 * n), G(2 * n), f(n), g(n); FOR(j, 2 * k) { move(P.begin() + 2 * n * j, P.begin() + 2 * n * j + 2 * n, F.begin()); move(Q.begin() + 2 * n * j, Q.begin() + 2 * n * j + 2 * n, G.begin()); FOR(i, n) { f[i] = W[i] * (F[2 * i] * G[2 * i + 1] - F[2 * i + 1] * G[2 * i]); g[i] = G[2 * i] * G[2 * i + 1]; } ntt(f, 1), ntt(g, 1); fill(f.begin() + n / 2, f.end(), mint(0)); fill(g.begin() + n / 2, g.end(), mint(0)); move(all(f), P.begin() + n * j); move(all(g), Q.begin() + n * j); } fill(P.begin() + 2 * n * k, P.end(), mint(0)); fill(Q.begin() + 2 * n * k, Q.end(), mint(0)); FOR(j, 4 * k) Q[n * j] = 0; n /= 2, k *= 2; } FOR(i, k) P[i] = P[2 * i]; P.resize(k); mint c = mint(1) / mint(k); for (auto& x: P) x *= c; ntt(P, 1); reverse(all(P)); P.resize(m + 1); return P; } // \sum_jwt[j][x^j]f^i を i=0,1,...,m template <typename mint> vc<mint> power_projection_0_garner(vc<mint> wt, vc<mint> f, int m) { assert(len(f) == len(wt) && f[0] == mint(0)); int n = 1; while (n < len(f)) n *= 2; f.resize(n), wt.resize(n); reverse(all(wt)); constexpr u32 p[] = {167772161, 469762049, 754974721}; using mint0 = modint<p[0]>; using mint1 = modint<p[1]>; using mint2 = modint<p[2]>; vc<mint0> W0(2 * n); vc<mint1> W1(2 * n); vc<mint2> W2(2 * n); { // bit reverse order vc<int> btr(2 * n); int log = topbit(2 * n); FOR(i, 2 * n) { btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (log - 1)); } { int t = mint0::ntt_info().fi; mint0 r = mint0::ntt_info().se; mint0 dw = r.inverse().pow((1 << t) / (4 * n)); mint0 w = 1; for (auto& i: btr) { W0[i] = w, w *= dw; } } { int t = mint1::ntt_info().fi; mint1 r = mint1::ntt_info().se; mint1 dw = r.inverse().pow((1 << t) / (4 * n)); mint1 w = 1; for (auto& i: btr) { W1[i] = w, w *= dw; } } { int t = mint2::ntt_info().fi; mint2 r = mint2::ntt_info().se; mint2 dw = r.inverse().pow((1 << t) / (4 * n)); mint2 w = 1; for (auto& i: btr) { W2[i] = w, w *= dw; } } } int k = 1; vc<mint> P(2 * n), Q(2 * n); FOR(i, n) P[i] = wt[i], Q[i] = -f[i]; while (n > 1) { vc<mint0> P0(4 * n * k), Q0(4 * n * k); vc<mint1> P1(4 * n * k), Q1(4 * n * k); vc<mint2> P2(4 * n * k), Q2(4 * n * k); FOR(i, 2 * n * k) P0[i] = P[i].val, Q0[i] = Q[i].val; FOR(i, 2 * n * k) P1[i] = P[i].val, Q1[i] = Q[i].val; FOR(i, 2 * n * k) P2[i] = P[i].val, Q2[i] = Q[i].val; Q0[2 * n * k] = 1, Q1[2 * n * k] = 1, Q2[2 * n * k] = 1; ntt(P0, 0), ntt(Q0, 0), ntt(P1, 0), ntt(Q1, 0), ntt(P2, 0), ntt(Q2, 0); FOR(i, 2 * n * k) { P0[i] = inv<mint0>(2) * W0[i] * (P0[2 * i] * Q0[2 * i + 1] - P0[2 * i + 1] * Q0[2 * i]); Q0[i] = Q0[2 * i] * Q0[2 * i + 1]; P1[i] = inv<mint1>(2) * W1[i] * (P1[2 * i] * Q1[2 * i + 1] - P1[2 * i + 1] * Q1[2 * i]); Q1[i] = Q1[2 * i] * Q1[2 * i + 1]; P2[i] = inv<mint2>(2) * W2[i] * (P2[2 * i] * Q2[2 * i + 1] - P2[2 * i + 1] * Q2[2 * i]); Q2[i] = Q2[2 * i] * Q2[2 * i + 1]; } P0.resize(2 * n * k), Q0.resize(2 * n * k); P1.resize(2 * n * k), Q1.resize(2 * n * k); P2.resize(2 * n * k), Q2.resize(2 * n * k); ntt(P0, 1), ntt(Q0, 1), ntt(P1, 1), ntt(Q1, 1), ntt(P2, 1), ntt(Q2, 1); constexpr i128 K = u128(p[0]) * p[1] * p[2]; auto get = [&](mint0 a, mint1 b, mint2 c) -> mint { i128 x = CRT3<u128, p[0], p[1], p[2]>(a.val, b.val, c.val); i128 y = K - x; return (x < y ? mint(x) : -mint(y)); }; fill(all(P), mint(0)); fill(all(Q), mint(0)); FOR(j, 2 * k) FOR(i, n / 2) { int k = n * j + i; P[k] = get(P0[k], P1[k], P2[k]); Q[k] = get(Q0[k], Q1[k], Q2[k]); } Q[0] = 0; n /= 2, k *= 2; } vc<mint> F(k); FOR(i, k) F[i] = P[2 * i]; reverse(all(F)); F.resize(m + 1); return F; } // \sum_j[x^j]f^i を i=0,1,...,m template <typename mint> vc<mint> power_projection(vc<mint> wt, vc<mint> f, int m) { assert(len(f) == len(wt)); if (f.empty()) { return vc<mint>(m + 1, mint(0)); } if (f[0] != mint(0)) { mint c = f[0]; f[0] = 0; vc<mint> A = power_projection(wt, f, m); FOR(p, m + 1) A[p] *= fact_inv<mint>(p); vc<mint> B(m + 1); mint pow = 1; FOR(q, m + 1) B[q] = pow * fact_inv<mint>(q), pow *= c; A = convolution<mint>(A, B); A.resize(m + 1); FOR(i, m + 1) A[i] *= fact<mint>(i); return A; } if (mint::can_ntt()) { return power_projection_0_ntt(wt, f, m); } return power_projection_0_garner(wt, f, m); }

#line 2 "poly/power_projection.hpp" #line 2 "mod/modint_common.hpp" struct has_mod_impl { template <class T> static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{}); template <class T> static auto check(...) -> std::false_type; }; template <class T> class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {}; template <typename mint> mint inv(int n) { static const int mod = mint::get_mod(); static vector<mint> dat = {0, 1}; assert(0 <= n); if (n >= mod) n %= mod; while (len(dat) <= n) { int k = len(dat); int q = (mod + k - 1) / k; dat.eb(dat[k * q - mod] * mint::raw(q)); } return dat[n]; } template <typename mint> mint fact(int n) { static const int mod = mint::get_mod(); assert(0 <= n && n < mod); static vector<mint> dat = {1, 1}; while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat))); return dat[n]; } template <typename mint> mint fact_inv(int n) { static vector<mint> dat = {1, 1}; if (n < 0) return mint(0); while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat))); return dat[n]; } template <class mint, class... Ts> mint fact_invs(Ts... xs) { return (mint(1) * ... * fact_inv<mint>(xs)); } template <typename mint, class Head, class... Tail> mint multinomial(Head &&head, Tail &&... tail) { return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...); } template <typename mint> mint C_dense(int n, int k) { static vvc<mint> C; static int H = 0, W = 0; auto calc = [&](int i, int j) -> mint { if (i == 0) return (j == 0 ? mint(1) : mint(0)); return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0); }; if (W <= k) { FOR(i, H) { C[i].resize(k + 1); FOR(j, W, k + 1) { C[i][j] = calc(i, j); } } W = k + 1; } if (H <= n) { C.resize(n + 1); FOR(i, H, n + 1) { C[i].resize(W); FOR(j, W) { C[i][j] = calc(i, j); } } H = n + 1; } return C[n][k]; } template <typename mint, bool large = false, bool dense = false> mint C(ll n, ll k) { assert(n >= 0); if (k < 0 || n < k) return 0; if constexpr (dense) return C_dense<mint>(n, k); if constexpr (!large) return multinomial<mint>(n, k, n - k); k = min(k, n - k); mint x(1); FOR(i, k) x *= mint(n - i); return x * fact_inv<mint>(k); } template <typename mint, bool large = false> mint C_inv(ll n, ll k) { assert(n >= 0); assert(0 <= k && k <= n); if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k); return mint(1) / C<mint, 1>(n, k); } // [x^d](1-x)^{-n} template <typename mint, bool large = false, bool dense = false> mint C_negative(ll n, ll d) { assert(n >= 0); if (d < 0) return mint(0); if (n == 0) { return (d == 0 ? mint(1) : mint(0)); } return C<mint, large, dense>(n + d - 1, d); } #line 3 "mod/modint.hpp" template <int mod> struct modint { static constexpr u32 umod = u32(mod); static_assert(umod < u32(1) << 31); u32 val; static modint raw(u32 v) { modint x; x.val = v; return x; } constexpr modint() : val(0) {} constexpr modint(u32 x) : val(x % umod) {} constexpr modint(u64 x) : val(x % umod) {} constexpr modint(u128 x) : val(x % umod) {} constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){}; constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){}; constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){}; bool operator<(const modint &other) const { return val < other.val; } modint &operator+=(const modint &p) { if ((val += p.val) >= umod) val -= umod; return *this; } modint &operator-=(const modint &p) { if ((val += umod - p.val) >= umod) val -= umod; return *this; } modint &operator*=(const modint &p) { val = u64(val) * p.val % umod; return *this; } modint &operator/=(const modint &p) { *this *= p.inverse(); return *this; } modint operator-() const { return modint::raw(val ? mod - val : u32(0)); } modint operator+(const modint &p) const { return modint(*this) += p; } modint operator-(const modint &p) const { return modint(*this) -= p; } modint operator*(const modint &p) const { return modint(*this) *= p; } modint operator/(const modint &p) const { return modint(*this) /= p; } bool operator==(const modint &p) const { return val == p.val; } bool operator!=(const modint &p) const { return val != p.val; } modint inverse() const { int a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } return modint(u); } modint pow(ll n) const { assert(n >= 0); modint ret(1), mul(val); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } static constexpr int get_mod() { return mod; } // (n, r), r は 1 の 2^n 乗根 static constexpr pair<int, int> ntt_info() { if (mod == 120586241) return {20, 74066978}; if (mod == 167772161) return {25, 17}; if (mod == 469762049) return {26, 30}; if (mod == 754974721) return {24, 362}; if (mod == 880803841) return {23, 211}; if (mod == 943718401) return {22, 663003469}; if (mod == 998244353) return {23, 31}; if (mod == 1004535809) return {21, 836905998}; if (mod == 1045430273) return {20, 363}; if (mod == 1051721729) return {20, 330}; if (mod == 1053818881) return {20, 2789}; return {-1, -1}; } static constexpr bool can_ntt() { return ntt_info().fi != -1; } }; #ifdef FASTIO template <int mod> void rd(modint<mod> &x) { fastio::rd(x.val); x.val %= mod; // assert(0 <= x.val && x.val < mod); } template <int mod> void wt(modint<mod> x) { fastio::wt(x.val); } #endif using modint107 = modint<1000000007>; using modint998 = modint<998244353>; #line 2 "mod/mod_inv.hpp" // long でも大丈夫 // (val * x - 1) が mod の倍数になるようにする // 特に mod=0 なら x=0 が満たす ll mod_inv(ll val, ll mod) { if (mod == 0) return 0; mod = abs(mod); val %= mod; if (val < 0) val += mod; ll a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } if (u < 0) u += mod; return u; } #line 2 "mod/crt3.hpp" constexpr u32 mod_pow_constexpr(u64 a, u64 n, u32 mod) { a %= mod; u64 res = 1; FOR(32) { if (n & 1) res = res * a % mod; a = a * a % mod, n /= 2; } return res; } template <typename T, u32 p0, u32 p1> T CRT2(u64 a0, u64 a1) { static_assert(p0 < p1); static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1); u64 c = (a1 - a0 + p1) * x0_1 % p1; return a0 + c * p0; } template <typename T, u32 p0, u32 p1, u32 p2> T CRT3(u64 a0, u64 a1, u64 a2) { static_assert(p0 < p1 && p1 < p2); static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); static constexpr u64 p01 = u64(p0) * p1; u64 c = (a1 - a0 + p1) * x1 % p1; u64 ans_1 = a0 + c * p0; c = (a2 - ans_1 % p2 + p2) * x2 % p2; return T(ans_1) + T(c) * T(p01); } template <typename T, u32 p0, u32 p1, u32 p2, u32 p3, u32 p4> T CRT5(u64 a0, u64 a1, u64 a2, u64 a3, u64 a4) { static_assert(p0 < p1 && p1 < p2 && p2 < p3 && p3 < p4); static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3); static constexpr u64 x4 = mod_pow_constexpr(u64(p0) * p1 % p4 * p2 % p4 * p3 % p4, p4 - 2, p4); static constexpr u64 p01 = u64(p0) * p1; static constexpr u64 p23 = u64(p2) * p3; u64 c = (a1 - a0 + p1) * x1 % p1; u64 ans_1 = a0 + c * p0; c = (a2 - ans_1 % p2 + p2) * x2 % p2; u128 ans_2 = ans_1 + c * static_cast<u128>(p01); c = static_cast<u64>(a3 - ans_2 % p3 + p3) * x3 % p3; u128 ans_3 = ans_2 + static_cast<u128>(c * p2) * p01; c = static_cast<u64>(a4 - ans_3 % p4 + p4) * x4 % p4; return T(ans_3) + T(c) * T(p01) * T(p23); } #line 2 "poly/convolution_naive.hpp" template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr> vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) { int n = int(a.size()), m = int(b.size()); if (n > m) return convolution_naive<T>(b, a); if (n == 0) return {}; vector<T> ans(n + m - 1); FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j]; return ans; } template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr> vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) { int n = int(a.size()), m = int(b.size()); if (n > m) return convolution_naive<T>(b, a); if (n == 0) return {}; vc<T> ans(n + m - 1); if (n <= 16 && (T::get_mod() < (1 << 30))) { for (int k = 0; k < n + m - 1; ++k) { int s = max(0, k - m + 1); int t = min(n, k + 1); u64 sm = 0; for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); } ans[k] = sm; } } else { for (int k = 0; k < n + m - 1; ++k) { int s = max(0, k - m + 1); int t = min(n, k + 1); u128 sm = 0; for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); } ans[k] = T::raw(sm % T::get_mod()); } } return ans; } #line 2 "poly/convolution_karatsuba.hpp" // 任意の環でできる template <typename T> vc<T> convolution_karatsuba(const vc<T>& f, const vc<T>& g) { const int thresh = 30; if (min(len(f), len(g)) <= thresh) return convolution_naive(f, g); int n = max(len(f), len(g)); int m = ceil(n, 2); vc<T> f1, f2, g1, g2; if (len(f) < m) f1 = f; if (len(f) >= m) f1 = {f.begin(), f.begin() + m}; if (len(f) >= m) f2 = {f.begin() + m, f.end()}; if (len(g) < m) g1 = g; if (len(g) >= m) g1 = {g.begin(), g.begin() + m}; if (len(g) >= m) g2 = {g.begin() + m, g.end()}; vc<T> a = convolution_karatsuba(f1, g1); vc<T> b = convolution_karatsuba(f2, g2); FOR(i, len(f2)) f1[i] += f2[i]; FOR(i, len(g2)) g1[i] += g2[i]; vc<T> c = convolution_karatsuba(f1, g1); vc<T> F(len(f) + len(g) - 1); assert(2 * m + len(b) <= len(F)); FOR(i, len(a)) F[i] += a[i], c[i] -= a[i]; FOR(i, len(b)) F[2 * m + i] += b[i], c[i] -= b[i]; if (c.back() == T(0)) c.pop_back(); FOR(i, len(c)) if (c[i] != T(0)) F[m + i] += c[i]; return F; } #line 2 "poly/ntt.hpp" template <class mint> void ntt(vector<mint>& a, bool inverse) { assert(mint::can_ntt()); const int rank2 = mint::ntt_info().fi; const int mod = mint::get_mod(); static array<mint, 30> root, iroot; static array<mint, 30> rate2, irate2; static array<mint, 30> rate3, irate3; assert(rank2 != -1 && len(a) <= (1 << max(0, rank2))); static bool prepared = 0; if (!prepared) { prepared = 1; root[rank2] = mint::ntt_info().se; iroot[rank2] = mint(1) / root[rank2]; FOR_R(i, rank2) { root[i] = root[i + 1] * root[i + 1]; iroot[i] = iroot[i + 1] * iroot[i + 1]; } mint prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 2; i++) { rate2[i] = root[i + 2] * prod; irate2[i] = iroot[i + 2] * iprod; prod *= iroot[i + 2]; iprod *= root[i + 2]; } prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 3; i++) { rate3[i] = root[i + 3] * prod; irate3[i] = iroot[i + 3] * iprod; prod *= iroot[i + 3]; iprod *= root[i + 3]; } } int n = int(a.size()); int h = topbit(n); assert(n == 1 << h); if (!inverse) { int len = 0; while (len < h) { if (h - len == 1) { int p = 1 << (h - len - 1); mint rot = 1; FOR(s, 1 << len) { int offset = s << (h - len); FOR(i, p) { auto l = a[i + offset]; auto r = a[i + offset + p] * rot; a[i + offset] = l + r; a[i + offset + p] = l - r; } rot *= rate2[topbit(~s & -~s)]; } len++; } else { int p = 1 << (h - len - 2); mint rot = 1, imag = root[2]; for (int s = 0; s < (1 << len); s++) { mint rot2 = rot * rot; mint rot3 = rot2 * rot; int offset = s << (h - len); for (int i = 0; i < p; i++) { u64 mod2 = u64(mod) * mod; u64 a0 = a[i + offset].val; u64 a1 = u64(a[i + offset + p].val) * rot.val; u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val; u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val; u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val; u64 na2 = mod2 - a2; a[i + offset] = a0 + a2 + a1 + a3; a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3)); a[i + offset + 2 * p] = a0 + na2 + a1na3imag; a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag); } rot *= rate3[topbit(~s & -~s)]; } len += 2; } } } else { mint coef = mint(1) / mint(len(a)); FOR(i, len(a)) a[i] *= coef; int len = h; while (len) { if (len == 1) { int p = 1 << (h - len); mint irot = 1; FOR(s, 1 << (len - 1)) { int offset = s << (h - len + 1); FOR(i, p) { u64 l = a[i + offset].val; u64 r = a[i + offset + p].val; a[i + offset] = l + r; a[i + offset + p] = (mod + l - r) * irot.val; } irot *= irate2[topbit(~s & -~s)]; } len--; } else { int p = 1 << (h - len); mint irot = 1, iimag = iroot[2]; FOR(s, (1 << (len - 2))) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; int offset = s << (h - len + 2); for (int i = 0; i < p; i++) { u64 a0 = a[i + offset + 0 * p].val; u64 a1 = a[i + offset + 1 * p].val; u64 a2 = a[i + offset + 2 * p].val; u64 a3 = a[i + offset + 3 * p].val; u64 x = (mod + a2 - a3) * iimag.val % mod; a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val; a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val; a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val; } irot *= irate3[topbit(~s & -~s)]; } len -= 2; } } } } #line 1 "poly/fft.hpp" namespace CFFT { using real = double; struct C { real x, y; C() : x(0), y(0) {} C(real x, real y) : x(x), y(y) {} inline C operator+(const C& c) const { return C(x + c.x, y + c.y); } inline C operator-(const C& c) const { return C(x - c.x, y - c.y); } inline C operator*(const C& c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); } inline C conj() const { return C(x, -y); } }; const real PI = acosl(-1); int base = 1; vector<C> rts = {{0, 0}, {1, 0}}; vector<int> rev = {0, 1}; void ensure_base(int nbase) { if (nbase <= base) return; rev.resize(1 << nbase); rts.resize(1 << nbase); for (int i = 0; i < (1 << nbase); i++) { rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } while (base < nbase) { real angle = PI * 2.0 / (1 << (base + 1)); for (int i = 1 << (base - 1); i < (1 << base); i++) { rts[i << 1] = rts[i]; real angle_i = angle * (2 * i + 1 - (1 << base)); rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i)); } ++base; } } void fft(vector<C>& a, int n) { assert((n & (n - 1)) == 0); int zeros = __builtin_ctz(n); ensure_base(zeros); int shift = base - zeros; for (int i = 0; i < n; i++) { if (i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); } } for (int k = 1; k < n; k <<= 1) { for (int i = 0; i < n; i += 2 * k) { for (int j = 0; j < k; j++) { C z = a[i + j + k] * rts[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } } // namespace CFFT #line 9 "poly/convolution.hpp" template <class mint> vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) { if (a.empty() || b.empty()) return {}; int n = int(a.size()), m = int(b.size()); int sz = 1; while (sz < n + m - 1) sz *= 2; // sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。 if ((n + m - 3) <= sz / 2) { auto a_last = a.back(), b_last = b.back(); a.pop_back(), b.pop_back(); auto c = convolution(a, b); c.resize(n + m - 1); c[n + m - 2] = a_last * b_last; FOR(i, len(a)) c[i + len(b)] += a[i] * b_last; FOR(i, len(b)) c[i + len(a)] += b[i] * a_last; return c; } a.resize(sz), b.resize(sz); bool same = a == b; ntt(a, 0); if (same) { b = a; } else { ntt(b, 0); } FOR(i, sz) a[i] *= b[i]; ntt(a, 1); a.resize(n + m - 1); return a; } template <typename mint> vector<mint> convolution_garner(const vector<mint>& a, const vector<mint>& b) { int n = len(a), m = len(b); if (!n || !m) return {}; static constexpr int p0 = 167772161; static constexpr int p1 = 469762049; static constexpr int p2 = 754974721; using mint0 = modint<p0>; using mint1 = modint<p1>; using mint2 = modint<p2>; vc<mint0> a0(n), b0(m); vc<mint1> a1(n), b1(m); vc<mint2> a2(n), b2(m); FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val; FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val; auto c0 = convolution_ntt<mint0>(a0, b0); auto c1 = convolution_ntt<mint1>(a1, b1); auto c2 = convolution_ntt<mint2>(a2, b2); vc<mint> c(len(c0)); FOR(i, n + m - 1) { c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val); } return c; } template <typename R> vc<double> convolution_fft(const vc<R>& a, const vc<R>& b) { using C = CFFT::C; int need = (int)a.size() + (int)b.size() - 1; int nbase = 1; while ((1 << nbase) < need) nbase++; CFFT::ensure_base(nbase); int sz = 1 << nbase; vector<C> fa(sz); for (int i = 0; i < sz; i++) { double x = (i < (int)a.size() ? a[i] : 0); double y = (i < (int)b.size() ? b[i] : 0); fa[i] = C(x, y); } CFFT::fft(fa, sz); C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0); for (int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r; fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r; fa[i] = z; } for (int i = 0; i < (sz >> 1); i++) { C A0 = (fa[i] + fa[i + (sz >> 1)]) * t; C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * CFFT::rts[(sz >> 1) + i]; fa[i] = A0 + A1 * s; } CFFT::fft(fa, sz >> 1); vector<double> ret(need); for (int i = 0; i < need; i++) { ret[i] = (i & 1 ? fa[i >> 1].y : fa[i >> 1].x); } return ret; } vector<ll> convolution(const vector<ll>& a, const vector<ll>& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (min(n, m) <= 2500) return convolution_naive(a, b); ll abs_sum_a = 0, abs_sum_b = 0; ll LIM = 1e15; FOR(i, n) abs_sum_a = min(LIM, abs_sum_a + abs(a[i])); FOR(i, m) abs_sum_b = min(LIM, abs_sum_b + abs(b[i])); if (i128(abs_sum_a) * abs_sum_b < 1e15) { vc<double> c = convolution_fft<ll>(a, b); vc<ll> res(len(c)); FOR(i, len(c)) res[i] = ll(floor(c[i] + .5)); return res; } static constexpr u32 MOD1 = 167772161; // 2^25 static constexpr u32 MOD2 = 469762049; // 2^26 static constexpr u32 MOD3 = 754974721; // 2^24 using mint1 = modint<MOD1>; using mint2 = modint<MOD2>; using mint3 = modint<MOD3>; vc<mint1> a1(n), b1(m); vc<mint2> a2(n), b2(m); vc<mint3> a3(n), b3(m); FOR(i, n) a1[i] = a[i], a2[i] = a[i], a3[i] = a[i]; FOR(i, m) b1[i] = b[i], b2[i] = b[i], b3[i] = b[i]; auto c1 = convolution_ntt<mint1>(a1, b1); auto c2 = convolution_ntt<mint2>(a2, b2); auto c3 = convolution_ntt<mint3>(a3, b3); u128 prod = u128(MOD1) * MOD2 * MOD3; vc<ll> res(n + m - 1); FOR(i, n + m - 1) { u128 x = CRT3<u128, MOD1, MOD2, MOD3>(c1[i].val, c2[i].val, c3[i].val); res[i] = (x < prod / 2 ? ll(x) : -ll(prod - x)); } return res; } template <typename mint> vc<mint> convolution(const vc<mint>& a, const vc<mint>& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (mint::can_ntt()) { if (min(n, m) <= 50) return convolution_karatsuba<mint>(a, b); return convolution_ntt(a, b); } if (min(n, m) <= 200) return convolution_karatsuba<mint>(a, b); return convolution_garner(a, b); } #line 4 "poly/power_projection.hpp" template <typename mint> vc<mint> power_projection_0_ntt(vc<mint> wt, vc<mint> f, int m) { assert(len(f) == len(wt) && f[0] == mint(0)); int n = 1; while (n < len(f)) n *= 2; for (auto& x: f) x = -x; f.resize(n), wt.resize(n); reverse(all(wt)); vc<mint>&P = wt, &Q = f; P.resize(4 * n), Q.resize(4 * n); vc<mint> W(n); { // bit reverse order vc<int> btr(n); int log = topbit(n); FOR(i, n) { btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (log - 1)); } int t = mint::ntt_info().fi; mint r = mint::ntt_info().se; mint dw = r.inverse().pow((1 << t) / (2 * n)); mint w = 1; for (auto& i: btr) { W[i] = w, w *= dw; } } int k = 1; while (n > 1) { /* FFT step 04.. -> 048c 15.. -> 159d .... -> 26ae .... -> 37bf */ auto doubling_y = [&](vc<mint>& A, int l, int r) -> void { mint z = W[k / 2].inverse(); vc<mint> f(k); FOR(i, l, r) { FOR(j, k) f[j] = A[2 * n * j + i]; ntt(f, 1); mint r = 1; FOR(j, 1, k) r *= z, f[j] *= r; ntt(f, 0); FOR(j, k) A[2 * n * (k + j) + i] = f[j]; } }; auto FFT_x = [&](vc<mint>& A, int l, int r) -> void { vc<mint> f(2 * n); FOR(j, l, r) { move(A.begin() + 2 * n * j, A.begin() + 2 * n * (j + 1), f.begin()); ntt(f, 0); move(all(f), A.begin() + 2 * n * j); } }; if (n <= k) { doubling_y(P, 0, n), doubling_y(Q, 1, n); FFT_x(P, 0, 2 * k), FFT_x(Q, 0, 2 * k); } else { FFT_x(P, 0, k), FFT_x(Q, 0, k); doubling_y(P, 0, 2 * n), doubling_y(Q, 0, 2 * n); } FOR(i, 2 * n * k) Q[i] += 1; FOR(i, 2 * n * k, 4 * n * k) Q[i] -= 1; /* 048c -> 0248???? 159d -> ....???? 26ae 37bf */ vc<mint> F(2 * n), G(2 * n), f(n), g(n); FOR(j, 2 * k) { move(P.begin() + 2 * n * j, P.begin() + 2 * n * j + 2 * n, F.begin()); move(Q.begin() + 2 * n * j, Q.begin() + 2 * n * j + 2 * n, G.begin()); FOR(i, n) { f[i] = W[i] * (F[2 * i] * G[2 * i + 1] - F[2 * i + 1] * G[2 * i]); g[i] = G[2 * i] * G[2 * i + 1]; } ntt(f, 1), ntt(g, 1); fill(f.begin() + n / 2, f.end(), mint(0)); fill(g.begin() + n / 2, g.end(), mint(0)); move(all(f), P.begin() + n * j); move(all(g), Q.begin() + n * j); } fill(P.begin() + 2 * n * k, P.end(), mint(0)); fill(Q.begin() + 2 * n * k, Q.end(), mint(0)); FOR(j, 4 * k) Q[n * j] = 0; n /= 2, k *= 2; } FOR(i, k) P[i] = P[2 * i]; P.resize(k); mint c = mint(1) / mint(k); for (auto& x: P) x *= c; ntt(P, 1); reverse(all(P)); P.resize(m + 1); return P; } // \sum_jwt[j][x^j]f^i を i=0,1,...,m template <typename mint> vc<mint> power_projection_0_garner(vc<mint> wt, vc<mint> f, int m) { assert(len(f) == len(wt) && f[0] == mint(0)); int n = 1; while (n < len(f)) n *= 2; f.resize(n), wt.resize(n); reverse(all(wt)); constexpr u32 p[] = {167772161, 469762049, 754974721}; using mint0 = modint<p[0]>; using mint1 = modint<p[1]>; using mint2 = modint<p[2]>; vc<mint0> W0(2 * n); vc<mint1> W1(2 * n); vc<mint2> W2(2 * n); { // bit reverse order vc<int> btr(2 * n); int log = topbit(2 * n); FOR(i, 2 * n) { btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (log - 1)); } { int t = mint0::ntt_info().fi; mint0 r = mint0::ntt_info().se; mint0 dw = r.inverse().pow((1 << t) / (4 * n)); mint0 w = 1; for (auto& i: btr) { W0[i] = w, w *= dw; } } { int t = mint1::ntt_info().fi; mint1 r = mint1::ntt_info().se; mint1 dw = r.inverse().pow((1 << t) / (4 * n)); mint1 w = 1; for (auto& i: btr) { W1[i] = w, w *= dw; } } { int t = mint2::ntt_info().fi; mint2 r = mint2::ntt_info().se; mint2 dw = r.inverse().pow((1 << t) / (4 * n)); mint2 w = 1; for (auto& i: btr) { W2[i] = w, w *= dw; } } } int k = 1; vc<mint> P(2 * n), Q(2 * n); FOR(i, n) P[i] = wt[i], Q[i] = -f[i]; while (n > 1) { vc<mint0> P0(4 * n * k), Q0(4 * n * k); vc<mint1> P1(4 * n * k), Q1(4 * n * k); vc<mint2> P2(4 * n * k), Q2(4 * n * k); FOR(i, 2 * n * k) P0[i] = P[i].val, Q0[i] = Q[i].val; FOR(i, 2 * n * k) P1[i] = P[i].val, Q1[i] = Q[i].val; FOR(i, 2 * n * k) P2[i] = P[i].val, Q2[i] = Q[i].val; Q0[2 * n * k] = 1, Q1[2 * n * k] = 1, Q2[2 * n * k] = 1; ntt(P0, 0), ntt(Q0, 0), ntt(P1, 0), ntt(Q1, 0), ntt(P2, 0), ntt(Q2, 0); FOR(i, 2 * n * k) { P0[i] = inv<mint0>(2) * W0[i] * (P0[2 * i] * Q0[2 * i + 1] - P0[2 * i + 1] * Q0[2 * i]); Q0[i] = Q0[2 * i] * Q0[2 * i + 1]; P1[i] = inv<mint1>(2) * W1[i] * (P1[2 * i] * Q1[2 * i + 1] - P1[2 * i + 1] * Q1[2 * i]); Q1[i] = Q1[2 * i] * Q1[2 * i + 1]; P2[i] = inv<mint2>(2) * W2[i] * (P2[2 * i] * Q2[2 * i + 1] - P2[2 * i + 1] * Q2[2 * i]); Q2[i] = Q2[2 * i] * Q2[2 * i + 1]; } P0.resize(2 * n * k), Q0.resize(2 * n * k); P1.resize(2 * n * k), Q1.resize(2 * n * k); P2.resize(2 * n * k), Q2.resize(2 * n * k); ntt(P0, 1), ntt(Q0, 1), ntt(P1, 1), ntt(Q1, 1), ntt(P2, 1), ntt(Q2, 1); constexpr i128 K = u128(p[0]) * p[1] * p[2]; auto get = [&](mint0 a, mint1 b, mint2 c) -> mint { i128 x = CRT3<u128, p[0], p[1], p[2]>(a.val, b.val, c.val); i128 y = K - x; return (x < y ? mint(x) : -mint(y)); }; fill(all(P), mint(0)); fill(all(Q), mint(0)); FOR(j, 2 * k) FOR(i, n / 2) { int k = n * j + i; P[k] = get(P0[k], P1[k], P2[k]); Q[k] = get(Q0[k], Q1[k], Q2[k]); } Q[0] = 0; n /= 2, k *= 2; } vc<mint> F(k); FOR(i, k) F[i] = P[2 * i]; reverse(all(F)); F.resize(m + 1); return F; } // \sum_j[x^j]f^i を i=0,1,...,m template <typename mint> vc<mint> power_projection(vc<mint> wt, vc<mint> f, int m) { assert(len(f) == len(wt)); if (f.empty()) { return vc<mint>(m + 1, mint(0)); } if (f[0] != mint(0)) { mint c = f[0]; f[0] = 0; vc<mint> A = power_projection(wt, f, m); FOR(p, m + 1) A[p] *= fact_inv<mint>(p); vc<mint> B(m + 1); mint pow = 1; FOR(q, m + 1) B[q] = pow * fact_inv<mint>(q), pow *= c; A = convolution<mint>(A, B); A.resize(m + 1); FOR(i, m + 1) A[i] *= fact<mint>(i); return A; } if (mint::can_ntt()) { return power_projection_0_ntt(wt, f, m); } return power_projection_0_garner(wt, f, m); }