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#include "poly/fps_sqrt.hpp"

#include "poly/fps_inv.hpp" #include "poly/fps_pow.hpp" #include "poly/count_terms.hpp" #include "mod/mod_sqrt.hpp" template <typename mint> vc<mint> fps_sqrt_dense(vc<mint>& f) { assert(f[0] == mint(1)); int n = len(f); vc<mint> R = {1}; while (len(R) < n) { int m = min(2 * int(len(R)), n); R.resize(m); vc<mint> tmp = {f.begin(), f.begin() + m}; tmp = convolution(tmp, fps_inv(R)); tmp.resize(m); FOR(i, m) R[i] += tmp[i]; mint c = mint(1) / mint(2); FOR(i, len(R)) R[i] *= c; } R.resize(n); return R; } template <typename mint> vc<mint> fps_sqrt_sparse(vc<mint>& f) { return fps_pow_1_sparse(f, inv<mint>(2)); } template <typename mint> vc<mint> fps_sqrt(vc<mint>& f) { if (count_terms(f) <= 200) return fps_sqrt_sparse(f); return fps_sqrt_dense(f); } template <typename mint> vc<mint> fps_sqrt_any(vc<mint>& f) { int n = len(f); int d = n; FOR_R(i, n) if (f[i] != 0) d = i; if (d == n) return f; if (d & 1) return {}; mint y = f[d]; mint x = mod_sqrt(y.val, mint::get_mod()); if (x * x != y) return {}; mint c = mint(1) / y; vc<mint> g(n - d); FOR(i, n - d) g[i] = f[d + i] * c; g = fps_sqrt(g); FOR(i, len(g)) g[i] *= x; g.resize(n); FOR_R(i, n) { if (i >= d / 2) g[i] = g[i - d / 2]; else g[i] = 0; } return g; }

#line 2 "poly/count_terms.hpp" template<typename mint> int count_terms(const vc<mint>& f){ int t = 0; FOR(i, len(f)) if(f[i] != mint(0)) ++t; return t; } #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 == 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 1 "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, u32 p2> T CRT3(u64 a0, u64 a1, u64 a2) { static_assert(p0 < p1 && p1 < p2); static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x01_2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); u64 c = (a1 - a0 + p1) * x0_1 % p1; u64 a = a0 + c * p0; c = (a2 - a % p2 + p2) * x01_2 % p2; return T(a) + T(c) * T(p0) * T(p1); } #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; 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++) { int x = (i < (int)a.size() ? a[i] : 0); int 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 unsigned long long MOD1 = 754974721; // 2^24 static constexpr unsigned long long MOD2 = 167772161; // 2^25 static constexpr unsigned long long MOD3 = 469762049; // 2^26 static constexpr unsigned long long M2M3 = MOD2 * MOD3; static constexpr unsigned long long M1M3 = MOD1 * MOD3; static constexpr unsigned long long M1M2 = MOD1 * MOD2; static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3; static const unsigned long long i1 = mod_inv(MOD2 * MOD3, MOD1); static const unsigned long long i2 = mod_inv(MOD1 * MOD3, MOD2); static const unsigned long long i3 = mod_inv(MOD1 * MOD2, MOD3); 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); vc<ll> c(n + m - 1); FOR(i, n + m - 1) { u64 x = 0; x += (c1[i].val * i1) % MOD1 * M2M3; x += (c2[i].val * i2) % MOD2 * M1M3; x += (c3[i].val * i3) % MOD3 * M1M2; ll diff = c1[i].val - ((long long)(x) % (long long)(MOD1)); if (diff < 0) diff += MOD1; static constexpr unsigned long long offset[5] = {0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3}; x -= offset[diff % 5]; c[i] = x; } return c; } 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/fps_inv.hpp" template <typename mint> vc<mint> fps_inv_sparse(const vc<mint>& f) { int N = len(f); vc<pair<int, mint>> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc<mint> g(N); mint g0 = mint(1) / f[0]; g[0] = g0; FOR(n, 1, N) { mint rhs = 0; for (auto&& [k, fk]: dat) { if (k > n) break; rhs -= fk * g[n - k]; } g[n] = rhs * g0; } return g; } template <typename mint> vc<mint> fps_inv_dense_ntt(const vc<mint>& F) { vc<mint> G = {mint(1) / F[0]}; ll N = len(F), n = 1; G.reserve(N); while (n < N) { vc<mint> f(2 * n), g(2 * n); FOR(i, min(N, 2 * n)) f[i] = F[i]; FOR(i, n) g[i] = G[i]; ntt(f, false), ntt(g, false); FOR(i, 2 * n) f[i] *= g[i]; ntt(f, true); FOR(i, n) f[i] = 0; ntt(f, false); FOR(i, 2 * n) f[i] *= g[i]; ntt(f, true); FOR(i, n, min(N, 2 * n)) G.eb(-f[i]); n *= 2; } return G; } template <typename mint> vc<mint> fps_inv_dense(const vc<mint>& F) { if (mint::can_ntt()) return fps_inv_dense_ntt(F); const int N = len(F); vc<mint> R = {mint(1) / F[0]}; vc<mint> p; int m = 1; while (m < N) { p = convolution(R, R); p.resize(m + m); vc<mint> f = {F.begin(), F.begin() + min(m + m, N)}; p = convolution(p, f); R.resize(m + m); FOR(i, m + m) R[i] = R[i] + R[i] - p[i]; m += m; } R.resize(N); return R; } template <typename mint> vc<mint> fps_inv(const vc<mint>& f) { assert(f[0] != mint(0)); int n = count_terms(f); int t = (mint::can_ntt() ? 160 : 820); return (n <= t ? fps_inv_sparse<mint>(f) : fps_inv_dense<mint>(f)); } #line 2 "poly/integrate.hpp" // 不定積分：integrate(f) // 定積分：integrate(f, L, R) template <typename mint> vc<mint> integrate(const vc<mint>& f) { vc<mint> g(len(f) + 1); FOR3(i, 1, len(g)) g[i] = f[i - 1] * inv<mint>(i); return g; } // 不定積分：integrate(f) // 定積分：integrate(f, L, R) template <typename mint> mint integrate(const vc<mint>& f, mint L, mint R) { mint I = 0; mint pow_L = 1, pow_R = 1; FOR(i, len(f)) { pow_L *= L, pow_R *= R; I += inv<mint>(i + 1) * f[i] * (pow_R - pow_L); } return I; } #line 2 "poly/differentiate.hpp" template <typename mint> vc<mint> differentiate(const vc<mint>& f) { if (len(f) <= 1) return {}; vc<mint> g(len(f) - 1); FOR(i, len(g)) g[i] = f[i + 1] * mint(i + 1); return g; } #line 6 "poly/fps_exp.hpp" template <typename mint> vc<mint> fps_exp_sparse(vc<mint>& f) { if (len(f) == 0) return {mint(1)}; assert(f[0] == 0); int N = len(f); // df を持たせる vc<pair<int, mint>> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i - 1, mint(i) * f[i]); vc<mint> F(N); F[0] = 1; FOR(n, 1, N) { mint rhs = 0; for (auto&& [k, fk]: dat) { if (k > n - 1) break; rhs += fk * F[n - 1 - k]; } F[n] = rhs * inv<mint>(n); } return F; } template <typename mint> vc<mint> fps_exp_dense(vc<mint>& h) { const int n = len(h); assert(n > 0 && h[0] == mint(0)); if (mint::can_ntt()) { vc<mint>& f = h; vc<mint> b = {1, (1 < n ? f[1] : 0)}; vc<mint> c = {1}, z1, z2 = {1, 1}; while (len(b) < n) { int m = len(b); auto y = b; y.resize(2 * m); ntt(y, 0); z1 = z2; vc<mint> z(m); FOR(i, m) z[i] = y[i] * z1[i]; ntt(z, 1); FOR(i, m / 2) z[i] = 0; ntt(z, 0); FOR(i, m) z[i] *= -z1[i]; ntt(z, 1); c.insert(c.end(), z.begin() + m / 2, z.end()); z2 = c; z2.resize(2 * m); ntt(z2, 0); vc<mint> x(f.begin(), f.begin() + m); FOR(i, len(x) - 1) x[i] = x[i + 1] * mint(i + 1); x.back() = 0; ntt(x, 0); FOR(i, m) x[i] *= y[i]; ntt(x, 1); FOR(i, m - 1) x[i] -= b[i + 1] * mint(i + 1); x.resize(m + m); FOR(i, m - 1) x[m + i] = x[i], x[i] = 0; ntt(x, 0); FOR(i, m + m) x[i] *= z2[i]; ntt(x, 1); FOR_R(i, len(x) - 1) x[i + 1] = x[i] * inv<mint>(i + 1); x[0] = 0; FOR3(i, m, min(n, m + m)) x[i] += f[i]; FOR(i, m) x[i] = 0; ntt(x, 0); FOR(i, m + m) x[i] *= y[i]; ntt(x, 1); b.insert(b.end(), x.begin() + m, x.end()); } b.resize(n); return b; } const int L = len(h); assert(L > 0 && h[0] == mint(0)); int LOG = 0; while (1 << LOG < L) ++LOG; h.resize(1 << LOG); auto dh = differentiate(h); vc<mint> f = {1}, g = {1}; int m = 1; vc<mint> p; FOR(LOG) { p = convolution(f, g); p.resize(m); p = convolution(p, g); p.resize(m); g.resize(m); FOR(i, m) g[i] += g[i] - p[i]; p = {dh.begin(), dh.begin() + m - 1}; p = convolution(f, p); p.resize(m + m - 1); FOR(i, m + m - 1) p[i] = -p[i]; FOR(i, m - 1) p[i] += mint(i + 1) * f[i + 1]; p = convolution(p, g); p.resize(m + m - 1); FOR(i, m - 1) p[i] += dh[i]; p = integrate(p); FOR(i, m + m) p[i] = h[i] - p[i]; p[0] += mint(1); f = convolution(f, p); f.resize(m + m); m += m; } f.resize(L); return f; } template <typename mint> vc<mint> fps_exp(vc<mint>& f) { int n = count_terms(f); int t = (mint::can_ntt() ? 320 : 3000); return (n <= t ? fps_exp_sparse<mint>(f) : fps_exp_dense<mint>(f)); } #line 2 "poly/fps_log.hpp" #line 5 "poly/fps_log.hpp" template <typename mint> vc<mint> fps_log_dense(const vc<mint>& f) { assert(f[0] == mint(1)); ll N = len(f); vc<mint> df = f; FOR(i, N) df[i] *= mint(i); df.erase(df.begin()); auto f_inv = fps_inv(f); auto g = convolution(df, f_inv); g.resize(N - 1); g.insert(g.begin(), 0); FOR(i, N) g[i] *= inv<mint>(i); return g; } template <typename mint> vc<mint> fps_log_sparse(const vc<mint>& f) { int N = f.size(); vc<pair<int, mint>> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc<mint> F(N); vc<mint> g(N - 1); for (int n = 0; n < N - 1; ++n) { mint rhs = mint(n + 1) * f[n + 1]; for (auto&& [i, fi]: dat) { if (i > n) break; rhs -= fi * g[n - i]; } g[n] = rhs; F[n + 1] = rhs * inv<mint>(n + 1); } return F; } template <typename mint> vc<mint> fps_log(const vc<mint>& f) { assert(f[0] == mint(1)); int n = count_terms(f); int t = (mint::can_ntt() ? 200 : 1200); return (n <= t ? fps_log_sparse<mint>(f) : fps_log_dense<mint>(f)); } #line 5 "poly/fps_pow.hpp" // fps の k 乗を求める。k >= 0 の前提である。 // 定数項が 1 で、k が mint の場合には、fps_pow_1 を使うこと。 // ・dense な場合： log, exp を使う O(NlogN) // ・sparse な場合： O(NK) template <typename mint> vc<mint> fps_pow(const vc<mint>& f, ll k) { assert(0 <= k); int n = len(f); if (k == 0) { vc<mint> g(n); g[0] = mint(1); return g; } int d = n; FOR_R(i, n) if (f[i] != 0) d = i; // d * k >= n if (d >= ceil<ll>(n, k)) { vc<mint> g(n); return g; } ll off = d * k; mint c = f[d]; mint c_inv = mint(1) / mint(c); vc<mint> g(n - off); FOR(i, n - off) g[i] = f[d + i] * c_inv; g = fps_pow_1(g, mint(k)); vc<mint> h(n); c = c.pow(k); FOR(i, len(g)) h[off + i] = g[i] * c; return h; } template <typename mint> vc<mint> fps_pow_1_sparse(const vc<mint>& f, mint K) { int N = len(f); assert(N == 0 || f[0] == mint(1)); vc<pair<int, mint>> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc<mint> g(N); g[0] = 1; FOR(n, N - 1) { mint& x = g[n + 1]; for (auto&& [d, cf]: dat) { if (d > n + 1) break; mint t = cf * g[n - d + 1]; x += t * (K * mint(d) - mint(n - d + 1)); } x *= inv<mint>(n + 1); } return g; } template <typename mint> vc<mint> fps_pow_1_dense(const vc<mint>& f, mint K) { assert(f[0] == mint(1)); auto log_f = fps_log(f); FOR(i, len(f)) log_f[i] *= K; return fps_exp_dense(log_f); } template <typename mint> vc<mint> fps_pow_1(const vc<mint>& f, mint K) { int n = count_terms(f); int t = (mint::can_ntt() ? 100 : 1300); return (n <= t ? fps_pow_1_sparse(f, K) : fps_pow_1_dense(f, K)); } // f^e, sparse, O(NMK) template <typename mint> vvc<mint> fps_pow_1_sparse_2d(vvc<mint> f, mint n) { assert(f[0][0] == mint(1)); int N = len(f), M = len(f[0]); vv(mint, dp, N, M); dp[0] = fps_pow_1_sparse<mint>(f[0], n); vc<tuple<int, int, mint>> dat; FOR(i, N) FOR(j, M) { if ((i > 0 || j > 0) && f[i][j] != mint(0)) dat.eb(i, j, f[i][j]); } FOR(i, 1, N) { FOR(j, M) { // F = f^n, f dF = n df F // [x^{i-1}y^j] mint lhs = 0, rhs = 0; for (auto&& [a, b, c]: dat) { if (a < i && b <= j) lhs += dp[i - a][j - b] * mint(i - a); if (a <= i && b <= j) rhs += dp[i - a][j - b] * c * mint(a); } dp[i][j] = (n * rhs - lhs) * inv<mint>(i); } } return dp; } #line 2 "random/base.hpp" u64 RNG_64() { static uint64_t x_ = uint64_t(chrono::duration_cast<chrono::nanoseconds>( chrono::high_resolution_clock::now().time_since_epoch()) .count()) * 10150724397891781847ULL; x_ ^= x_ << 7; return x_ ^= x_ >> 9; } u64 RNG(u64 lim) { return RNG_64() % lim; } ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); } #line 2 "mod/mod_pow.hpp" #line 2 "mod/mongomery_modint.hpp" // odd mod. // x の代わりに rx を持つ template <int id, typename U1, typename U2> struct Mongomery_modint { using mint = Mongomery_modint; inline static U1 m, r, n2; static constexpr int W = numeric_limits<U1>::digits; static void set_mod(U1 mod) { assert(mod & 1 && mod <= U1(1) << (W - 2)); m = mod, n2 = -U2(m) % m, r = m; FOR(5) r *= 2 - m * r; r = -r; assert(r * m == U1(-1)); } static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; } U1 x; Mongomery_modint() : x(0) {} Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){}; U1 val() const { U1 y = reduce(x); return y >= m ? y - m : y; } mint &operator+=(mint y) { x = ((x += y.x) >= m ? x - m : x); return *this; } mint &operator-=(mint y) { x -= (x >= y.x ? y.x : y.x - m); return *this; } mint &operator*=(mint y) { x = reduce(U2(x) * y.x); return *this; } mint operator+(mint y) const { return mint(*this) += y; } mint operator-(mint y) const { return mint(*this) -= y; } mint operator*(mint y) const { return mint(*this) *= y; } bool operator==(mint y) const { return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x); } bool operator!=(mint y) const { return not operator==(y); } mint pow(ll n) const { assert(n >= 0); mint y = 1, z = *this; for (; n; n >>= 1, z *= z) if (n & 1) y *= z; return y; } }; template <int id> using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>; template <int id> using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>; #line 2 "mod/barrett.hpp" // https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp struct Barrett { u32 m; u64 im; explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {} u32 umod() const { return m; } u32 modulo(u64 z) { if (m == 1) return 0; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; return (z - y + (z < y ? m : 0)); } u64 floor(u64 z) { if (m == 1) return z; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; return (z < y ? x - 1 : x); } pair<u64, u32> divmod(u64 z) { if (m == 1) return {z, 0}; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; if (z < y) return {x - 1, z - y + m}; return {x, z - y}; } u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); } }; struct Barrett_64 { u128 mod, mh, ml; explicit Barrett_64(u64 mod = 1) : mod(mod) { u128 m = u128(-1) / mod; if (m * mod + mod == u128(0)) ++m; mh = m >> 64; ml = m & u64(-1); } u64 umod() const { return mod; } u64 modulo(u128 x) { u128 z = (x & u64(-1)) * ml; z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64); z = (x >> 64) * mh + (z >> 64); x -= z * mod; return x < mod ? x : x - mod; } u64 mul(u64 a, u64 b) { return modulo(u128(a) * b); } }; #line 5 "mod/mod_pow.hpp" u32 mod_pow(int a, ll n, int mod) { assert(n >= 0); a = ((a %= mod) < 0 ? a + mod : a); if ((mod & 1) && (mod < (1 << 30))) { using mint = Mongomery_modint_32<202311021>; mint::set_mod(mod); return mint(a).pow(n).val(); } Barrett bt(mod); int r = 1; while (n) { if (n & 1) r = bt.mul(r, a); a = bt.mul(a, a), n >>= 1; } return r; } u64 mod_pow_64(ll a, ll n, u64 mod) { assert(n >= 0); a = ((a %= mod) < 0 ? a + mod : a); if ((mod & 1) && (mod < (u64(1) << 62))) { using mint = Mongomery_modint_64<202311021>; mint::set_mod(mod); return mint(a).pow(n).val(); } Barrett_64 bt(mod); ll r = 1; while (n) { if (n & 1) r = bt.mul(r, a); a = bt.mul(a, a), n >>= 1; } return r; } #line 3 "mod/mod_sqrt.hpp" // p は素数. 解なしは -1. int mod_sqrt(int a, int p) { if (p == 2) return a; if (a == 0) return 0; int k = (p - 1) / 2; if (mod_pow(a, k, p) != 1) return -1; auto find = [&]() -> pi { while (1) { ll b = RNG(2, p); ll D = (b * b - a) % p; if (D == 0) return {b, D}; if (mod_pow(D, k, p) != 1) return {b, D}; } }; auto [b, D] = find(); if (D == 0) return b; ++k; // (b + sqrt(D))^k ll f0 = b, f1 = 1, g0 = 1, g1 = 0; while (k) { if (k & 1) { tie(g0, g1) = mp(f0 * g0 + D * f1 % p * g1, f1 * g0 + f0 * g1); g0 %= p, g1 %= p; } tie(f0, f1) = mp(f0 * f0 + D * f1 % p * f1, 2 * f0 * f1); f0 %= p, f1 %= p; k >>= 1; } if (g0 < 0) g0 += p; return g0; } // p は素数. 解なしは -1. ll mod_sqrt_64(ll a, ll p) { if (p == 2) return a; if (a == 0) return 0; ll k = (p - 1) / 2; if (mod_pow_64(a, k, p) != 1) return -1; auto find = [&]() -> pair<i128, i128> { while (1) { i128 b = RNG(2, p); i128 D = b * b - a; if (D == 0) return {b, D}; if (mod_pow_64(D, k, p) != 1) return {b, D}; } }; auto [b, D] = find(); if (D == 0) return b; ++k; // (b + sqrt(D))^k i128 f0 = b, f1 = 1, g0 = 1, g1 = 0; while (k) { if (k & 1) { tie(g0, g1) = mp(f0 * g0 + D * f1 % p * g1, f1 * g0 + f0 * g1); g0 %= p, g1 %= p; } tie(f0, f1) = mp(f0 * f0 + D * f1 % p * f1, 2 * f0 * f1); f0 %= p, f1 %= p; k >>= 1; } return g0; } #line 5 "poly/fps_sqrt.hpp" template <typename mint> vc<mint> fps_sqrt_dense(vc<mint>& f) { assert(f[0] == mint(1)); int n = len(f); vc<mint> R = {1}; while (len(R) < n) { int m = min(2 * int(len(R)), n); R.resize(m); vc<mint> tmp = {f.begin(), f.begin() + m}; tmp = convolution(tmp, fps_inv(R)); tmp.resize(m); FOR(i, m) R[i] += tmp[i]; mint c = mint(1) / mint(2); FOR(i, len(R)) R[i] *= c; } R.resize(n); return R; } template <typename mint> vc<mint> fps_sqrt_sparse(vc<mint>& f) { return fps_pow_1_sparse(f, inv<mint>(2)); } template <typename mint> vc<mint> fps_sqrt(vc<mint>& f) { if (count_terms(f) <= 200) return fps_sqrt_sparse(f); return fps_sqrt_dense(f); } template <typename mint> vc<mint> fps_sqrt_any(vc<mint>& f) { int n = len(f); int d = n; FOR_R(i, n) if (f[i] != 0) d = i; if (d == n) return f; if (d & 1) return {}; mint y = f[d]; mint x = mod_sqrt(y.val, mint::get_mod()); if (x * x != y) return {}; mint c = mint(1) / y; vc<mint> g(n - d); FOR(i, n - d) g[i] = f[d + i] * c; g = fps_sqrt(g); FOR(i, len(g)) g[i] *= x; g.resize(n); FOR_R(i, n) { if (i >= d / 2) g[i] = g[i - d / 2]; else g[i] = 0; } return g; }