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#define PROBLEM "https://judge.yosupo.jp/problem/aplusb" #include "my_template.hpp" #include "mod/modint.hpp" #include "graph/count/count_labeled_biconnected.hpp" using mint = modint998; void test() { vc<mint> F = count_labeled_biconnected<mint>(10); vi ANS = { 0, 0, 1, 1, 10, 238, 11368, 1014888, 166537616, 50680432112, 29107809374336, }; FOR(i, 11) assert(F[i] == mint(ANS[i])); FOR(i, 11) { mint a = count_labeled_biconnected_single<mint>(i); assert(a == mint(ANS[i])); } } void solve() { int a, b; cin >> a >> b; cout << a + b << '\n'; } signed main() { test(); solve(); return 0; }
#line 1 "test/1_mytest/count_labeled_biconnected.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/aplusb" #line 1 "my_template.hpp" #if defined(LOCAL) #include <my_template_compiled.hpp> #else // https://codeforces.com/blog/entry/96344 #pragma GCC optimize("Ofast,unroll-loops") // いまの CF だとこれ入れると動かない? // #pragma GCC target("avx2,popcnt") #include <bits/stdc++.h> using namespace std; using ll = long long; using u8 = uint8_t; using u16 = uint16_t; using u32 = uint32_t; using u64 = uint64_t; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template <class T> constexpr T infty = 0; template <> constexpr int infty<int> = 1'010'000'000; template <> constexpr ll infty<ll> = 2'020'000'000'000'000'000; template <> constexpr u32 infty<u32> = infty<int>; template <> constexpr u64 infty<u64> = infty<ll>; template <> constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000; template <> constexpr double infty<double> = infty<ll>; template <> constexpr long double infty<long double> = infty<ll>; using pi = pair<ll, ll>; using vi = vector<ll>; template <class T> using vc = vector<T>; template <class T> using vvc = vector<vc<T>>; template <class T> using vvvc = vector<vvc<T>>; template <class T> using vvvvc = vector<vvvc<T>>; template <class T> using vvvvvc = vector<vvvvc<T>>; template <class T> using pq = priority_queue<T>; template <class T> using pqg = priority_queue<T, vector<T>, greater<T>>; #define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__)) #define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } int popcnt_sgn(int x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(ll x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u64 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template <typename T> T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template <typename T> T ceil(T x, T y) { return floor(x + y - 1, y); } template <typename T> T bmod(T x, T y) { return x - y * floor(x, y); } template <typename T> pair<T, T> divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template <typename T, typename U> T SUM(const vector<U> &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template <typename T> T POP(deque<T> &que) { T a = que.front(); que.pop_front(); return a; } template <typename T> T POP(pq<T> &que) { T a = que.top(); que.pop(); return a; } template <typename T> T POP(pqg<T> &que) { T a = que.top(); que.pop(); return a; } template <typename T> T POP(vc<T> &que) { T a = que.back(); que.pop_back(); return a; } template <typename F> ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template <typename F> double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template <class T, class S> inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template <class T, class S> inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc<int> s_to_vi(const string &S, char first_char) { vc<int> A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template <typename T, typename U> vector<T> cumsum(vector<U> &A, int off = 1) { int N = A.size(); vector<T> B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template <typename T> vector<int> argsort(const vector<T> &A) { vector<int> ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template <typename T> vc<T> rearrange(const vc<T> &A, const vc<int> &I) { vc<T> B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } template <typename T, typename... Vectors> void concat(vc<T> &first, const Vectors &... others) { vc<T> &res = first; (res.insert(res.end(), others.begin(), others.end()), ...); } #endif #line 4 "test/1_mytest/count_labeled_biconnected.test.cpp" #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, 582313106}; if (mod == 1012924417) return {21, 368093570}; 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 1 "graph/count/count_labeled_undirected.hpp" // https://oeis.org/A006125 template <typename mint> vc<mint> count_labeled_undirected(int N) { vc<mint> F(N + 1); mint pow2 = 1; F[0] = 1; FOR(i, 1, N + 1) F[i] = F[i - 1] * pow2, pow2 += pow2; return F; } #line 2 "poly/fps_log.hpp" #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/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> T CRT4(u64 a0, u64 a1, u64 a2, u64 a3) { static_assert(p0 < p1 && p1 < p2 && p2 < p3); 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 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; u128 ans_2 = ans_1 + c * static_cast<u128>(p01); c = (a3 - ans_2 % p3 + p3) * x3 % p3; return T(ans_2) + T(c) * T(p01) * T(p2); } 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 8 "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; } vector<ll> convolution(vector<ll> a, 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 mi_a = MIN(a), mi_b = MIN(b); for (auto& x: a) x -= mi_a; for (auto& x: b) x -= mi_b; assert(MAX(a) * MAX(b) <= 1e18); auto Ac = cumsum<ll>(a), Bc = cumsum<ll>(b); vi res(n + m - 1); for (int k = 0; k < n + m - 1; ++k) { int s = max(0, k - m + 1); int t = min(n, k + 1); res[k] += (t - s) * mi_a * mi_b; res[k] += mi_a * (Bc[k - s + 1] - Bc[k - t + 1]); res[k] += mi_b * (Ac[t] - Ac[s]); } static constexpr u32 MOD1 = 1004535809; static constexpr u32 MOD2 = 1012924417; using mint1 = modint<MOD1>; using mint2 = modint<MOD2>; vc<mint1> a1(n), b1(m); vc<mint2> a2(n), b2(m); FOR(i, n) a1[i] = a[i], a2[i] = a[i]; FOR(i, m) b1[i] = b[i], b2[i] = b[i]; auto c1 = convolution_ntt<mint1>(a1, b1); auto c2 = convolution_ntt<mint2>(a2, b2); FOR(i, n + m - 1) { res[i] += CRT2<u64, MOD1, MOD2>(c1[i].val, c2[i].val); } 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/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 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 3 "graph/count/count_labeled_connected.hpp" // https://oeis.org/A001187 template <typename mint> vc<mint> count_labeled_connected(int N) { vc<mint> F = count_labeled_undirected<mint>(N); FOR(i, N + 1) F[i] *= fact_inv<mint>(i); F = fps_log(F); FOR(i, N + 1) F[i] *= fact<mint>(i); return F; } #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 2 "poly/composition.hpp" #line 2 "poly/poly_taylor_shift.hpp" #line 2 "nt/primetable.hpp" template <typename T = int> vc<T> primetable(int LIM) { ++LIM; const int S = 32768; static int done = 2; static vc<T> primes = {2}, sieve(S + 1); if (done < LIM) { done = LIM; primes = {2}, sieve.assign(S + 1, 0); const int R = LIM / 2; primes.reserve(int(LIM / log(LIM) * 1.1)); vc<pair<int, int>> cp; for (int i = 3; i <= S; i += 2) { if (!sieve[i]) { cp.eb(i, i * i / 2); for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1; } } for (int L = 1; L <= R; L += S) { array<bool, S> block{}; for (auto& [p, idx]: cp) for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1; FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1); } } int k = LB(primes, LIM + 1); return {primes.begin(), primes.begin() + k}; } #line 3 "mod/powertable.hpp" // a^0, ..., a^N template <typename mint> vc<mint> powertable_1(mint a, ll N) { // table of a^i vc<mint> f(N + 1, 1); FOR(i, N) f[i + 1] = a * f[i]; return f; } // 0^e, ..., N^e template <typename mint> vc<mint> powertable_2(ll e, ll N) { auto primes = primetable(N); vc<mint> f(N + 1, 1); f[0] = mint(0).pow(e); for (auto&& p: primes) { if (p > N) break; mint xp = mint(p).pow(e); ll pp = p; while (pp <= N) { ll i = pp; while (i <= N) { f[i] *= xp; i += pp; } pp *= p; } } return f; } #line 5 "poly/poly_taylor_shift.hpp" // f(x) -> f(x+c) template <typename mint> vc<mint> poly_taylor_shift(vc<mint> f, mint c) { if (c == mint(0)) return f; ll N = len(f); FOR(i, N) f[i] *= fact<mint>(i); auto b = powertable_1<mint>(c, N); FOR(i, N) b[i] *= fact_inv<mint>(i); reverse(all(f)); f = convolution(f, b); f.resize(N); reverse(all(f)); FOR(i, N) f[i] *= fact_inv<mint>(i); return f; } #line 2 "poly/transposed_ntt.hpp" template <class mint> void transposed_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 = h; while (len > 0) { if (len == 1) { int p = 1 << (h - len); mint rot = 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) * rot.val; } rot *= rate2[topbit(~s & -~s)]; } len--; } else { int p = 1 << (h - len); mint rot = 1, imag = root[2]; FOR(s, (1 << (len - 2))) { int offset = s << (h - len + 2); mint rot2 = rot * rot; mint rot3 = rot2 * rot; 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) * imag.val % mod; a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + mod - a1 + x) * rot.val; a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * rot2.val; a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * rot3.val; } rot *= rate3[topbit(~s & -~s)]; } len -= 2; } } } else { mint coef = mint(1) / mint(len(a)); FOR(i, len(a)) a[i] *= coef; int len = 0; while (len < h) { if (len == h - 1) { int p = 1 << (h - len - 1); mint irot = 1; FOR(s, 1 << len) { int offset = s << (h - len); FOR(i, p) { auto l = a[i + offset]; auto r = a[i + offset + p] * irot; a[i + offset] = l + r; a[i + offset + p] = l - r; } irot *= irate2[topbit(~s & -~s)]; } len++; } else { int p = 1 << (h - len - 2); mint irot = 1, iimag = iroot[2]; for (int s = 0; s < (1 << len); s++) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; 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) * irot.val; u64 a2 = u64(a[i + offset + 2 * p].val) * irot2.val; u64 a3 = u64(a[i + offset + 3 * p].val) * irot3.val; u64 a1na3imag = (a1 + mod2 - a3) % mod * iimag.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); } irot *= irate3[topbit(~s & -~s)]; } len += 2; } } } } #line 6 "poly/composition.hpp" template <typename mint> vc<mint> composition_old(vc<mint>& Q, vc<mint>& P) { int n = len(P); assert(len(P) == len(Q)); int k = 1; while (k * k < n) ++k; // compute powers of P vv(mint, pow1, k + 1); pow1[0] = {1}; pow1[1] = P; FOR3(i, 2, k + 1) { pow1[i] = convolution(pow1[i - 1], pow1[1]); pow1[i].resize(n); } vv(mint, pow2, k + 1); pow2[0] = {1}; pow2[1] = pow1[k]; FOR3(i, 2, k + 1) { pow2[i] = convolution(pow2[i - 1], pow2[1]); pow2[i].resize(n); } vc<mint> ANS(n); FOR(i, k + 1) { vc<mint> f(n); FOR(j, k) { if (k * i + j < len(Q)) { mint coef = Q[k * i + j]; FOR(d, len(pow1[j])) f[d] += pow1[j][d] * coef; } } f = convolution(f, pow2[i]); f.resize(n); FOR(d, n) ANS[d] += f[d]; } return ANS; } // f(g(x)), O(Nlog^2N) template <typename mint> vc<mint> composition_0_ntt(vc<mint> f, vc<mint> g) { assert(len(f) == len(g)); if (f.empty()) return {}; int n0 = len(f); int n = 1; while (n < len(f)) n *= 2; f.resize(n), g.resize(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; } } auto rec = [&](auto& rec, int n, int k, vc<mint>& Q) -> vc<mint> { if (n == 1) { reverse(all(f)); transposed_ntt(f, 1); mint c = mint(1) / mint(k); for (auto& x: f) x *= c; vc<mint> p(4 * k); FOR(i, k) p[2 * i] = f[i]; return p; } auto doubling_y = [&](vc<mint>& A, int l, int r, bool t) -> void { mint z = W[k / 2].inverse(); vc<mint> f(k); if (!t) { 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]; } } else { FOR(i, l, r) { FOR(j, k) f[j] = A[2 * n * (k + j) + i]; transposed_ntt(f, 0); mint r = 1; FOR(j, 1, k) r *= z, f[j] *= r; transposed_ntt(f, 1); FOR(j, k) A[2 * n * j + i] += f[j]; } } }; auto FFT_x = [&](vc<mint>& A, int l, int r, bool t) -> void { vc<mint> f(2 * n); if (!t) { 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); } } else { FOR(j, l, r) { move(A.begin() + 2 * n * j, A.begin() + 2 * n * (j + 1), f.begin()); transposed_ntt(f, 0); move(all(f), A.begin() + 2 * n * j); } } }; if (n <= k) doubling_y(Q, 1, n, 0), FFT_x(Q, 0, 2 * k, 0); if (n > k) FFT_x(Q, 0, k, 0), doubling_y(Q, 0, 2 * n, 0); FOR(i, 2 * n * k) Q[i] += 1; FOR(i, 2 * n * k, 4 * n * k) Q[i] -= 1; vc<mint> nxt_Q(4 * n * k); vc<mint> F(2 * n), G(2 * n), f(n), g(n); FOR(j, 2 * k) { move(Q.begin() + 2 * n * j, Q.begin() + 2 * n * j + 2 * n, G.begin()); FOR(i, n) { g[i] = G[2 * i] * G[2 * i + 1]; } ntt(g, 1); move(g.begin(), g.begin() + n / 2, nxt_Q.begin() + n * j); } FOR(j, 4 * k) nxt_Q[n * j] = 0; vc<mint> p = rec(rec, n / 2, k * 2, nxt_Q); FOR_R(j, 2 * k) { move(p.begin() + n * j, p.begin() + n * j + n / 2, f.begin()); move(Q.begin() + 2 * n * j, Q.begin() + 2 * n * j + 2 * n, G.begin()); fill(f.begin() + n / 2, f.end(), mint(0)); transposed_ntt(f, 1); FOR(i, n) { f[i] *= W[i]; F[2 * i] = G[2 * i + 1] * f[i], F[2 * i + 1] = -G[2 * i] * f[i]; } move(F.begin(), F.end(), p.begin() + 2 * n * j); } if (n <= k) FFT_x(p, 0, 2 * k, 1), doubling_y(p, 0, n, 1); if (n > k) doubling_y(p, 0, 2 * n, 1), FFT_x(p, 0, k, 1); return p; }; vc<mint> Q(4 * n); FOR(i, n) Q[i] = -g[i]; vc<mint> p = rec(rec, n, 1, Q); p.resize(n); reverse(all(p)); p.resize(n0); return p; } template <typename mint> vc<mint> composition_0_garner(vc<mint> f, vc<mint> g) { constexpr u32 ps[] = {167772161, 469762049, 754974721}; using mint0 = modint<ps[0]>; using mint1 = modint<ps[1]>; using mint2 = modint<ps[2]>; auto rec = [&](auto& rec, int n, int k, vc<mint> Q) -> vc<mint> { if (n == 1) { vc<mint> p(2 * k); reverse(all(f)); FOR(i, k) p[2 * i] = f[i]; return p; } vc<mint0> Q0(4 * n * k), R0(4 * n * k), p0(4 * n * k); vc<mint1> Q1(4 * n * k), R1(4 * n * k), p1(4 * n * k); vc<mint2> Q2(4 * n * k), R2(4 * n * k), p2(4 * n * k); FOR(i, 2 * n * k) { Q0[i] = Q[i].val, R0[i] = (i % 2 == 0 ? Q[i].val : (-Q[i]).val); Q1[i] = Q[i].val, R1[i] = (i % 2 == 0 ? Q[i].val : (-Q[i]).val); Q2[i] = Q[i].val, R2[i] = (i % 2 == 0 ? Q[i].val : (-Q[i]).val); } ntt(Q0, 0), ntt(Q1, 0), ntt(Q2, 0), ntt(R0, 0), ntt(R1, 0), ntt(R2, 0); FOR(i, 4 * n * k) Q0[i] *= R0[i], Q1[i] *= R1[i], Q2[i] *= R2[i]; ntt(Q0, 1), ntt(Q1, 1), ntt(Q2, 1); vc<mint> QQ(4 * n * k); FOR(i, 4 * n * k) { QQ[i] = CRT3<mint, ps[0], ps[1], ps[2]>(Q0[i].val, Q1[i].val, Q2[i].val); } FOR(i, 0, 2 * n * k, 2) { QQ[2 * n * k + i] += Q[i] + Q[i]; } vc<mint> nxt_Q(2 * n * k); FOR(j, 2 * k) FOR(i, n / 2) { nxt_Q[n * j + i] = QQ[(2 * n) * j + (2 * i + 0)]; } vc<mint> nxt_p = rec(rec, n / 2, k * 2, nxt_Q); vc<mint> pq(4 * n * k); FOR(j, 2 * k) FOR(i, n / 2) { pq[(2 * n) * j + (2 * i + 1)] += nxt_p[n * j + i]; } vc<mint> p(2 * n * k); FOR(i, 2 * n * k) { p[i] += pq[2 * n * k + i]; } FOR(i, 4 * n * k) { p0[i] += pq[i].val, p1[i] += pq[i].val, p2[i] += pq[i].val; } transposed_ntt(p0, 1), transposed_ntt(p1, 1), transposed_ntt(p2, 1); FOR(i, 4 * n * k) p0[i] *= R0[i], p1[i] *= R1[i], p2[i] *= R2[i]; transposed_ntt(p0, 0), transposed_ntt(p1, 0), transposed_ntt(p2, 0); FOR(i, 2 * n * k) { p[i] += CRT3<mint, ps[0], ps[1], ps[2]>(p0[i].val, p1[i].val, p2[i].val); } return p; }; assert(len(f) == len(g)); int n = 1; while (n < len(f)) n *= 2; int out_len = len(f); f.resize(n), g.resize(n); int k = 1; vc<mint> Q(2 * n); FOR(i, n) Q[i] = -g[i]; vc<mint> p = rec(rec, n, k, Q); vc<mint> output(n); FOR(i, n) output[i] = p[i]; reverse(all(output)); output.resize(out_len); return output; } template <typename mint> vc<mint> composition(vc<mint> f, vc<mint> g) { assert(len(f) == len(g)); if (f.empty()) return {}; // [x^0]g=0 に帰着しておく if (g[0] != mint(0)) { f = poly_taylor_shift<mint>(f, g[0]); g[0] = 0; } if (mint::can_ntt()) { return composition_0_ntt(f, g); } return composition_0_garner(f, g); } #line 2 "poly/fps_div.hpp" #line 5 "poly/fps_div.hpp" // f/g. f の長さで出力される. template <typename mint, bool SPARSE = false> vc<mint> fps_div(vc<mint> f, vc<mint> g) { if (SPARSE || count_terms(g) < 200) return fps_div_sparse(f, g); int n = len(f); g.resize(n); g = fps_inv<mint>(g); f = convolution(f, g); f.resize(n); return f; } // f/g ただし g は sparse template <typename mint> vc<mint> fps_div_sparse(vc<mint> f, vc<mint>& g) { if (g[0] != mint(1)) { mint cf = g[0].inverse(); for (auto&& x: f) x *= cf; for (auto&& x: g) x *= cf; } vc<pair<int, mint>> dat; FOR(i, 1, len(g)) if (g[i] != mint(0)) dat.eb(i, -g[i]); FOR(i, len(f)) { for (auto&& [j, x]: dat) { if (i >= j) f[i] += x * f[i - j]; } } return 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 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 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 "poly/power_projection.hpp" #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); } #line 6 "poly/compositional_inverse.hpp" // O(N^2) template <typename mint> vc<mint> compositional_inverse_old(const vc<mint>& F) { const int N = len(F); if (N == 0) return {}; assert(F[0] == mint(0)); if (N == 1) return F; assert(F[0] == mint(0) && F[1] != mint(0)); vc<mint> DF = differentiate(F); vc<mint> G(2); G[1] = mint(1) / F[1]; while (len(G) < N) { // G:= G(x)-(F(G(x))-x)/DF(G(x)) int n = len(G); vc<mint> G1, G2; { vc<mint> FF(2 * n), GG(2 * n), DFF(n); FOR(i, min<int>(len(F), 2 * n)) FF[i] = F[i]; FOR(i, min<int>(len(DF), n)) DFF[i] = DF[i]; FOR(i, n) GG[i] = G[i]; G1 = composition(FF, GG); G2 = composition(DFF, G); } G1 = {G1.begin() + n, G1.end()}; G1 = fps_div(G1, G2); G.resize(2 * n); FOR(i, n) G[n + i] -= G1[i]; } G.resize(N); return G; } template <typename mint> vc<mint> compositional_inverse(vc<mint> f) { const int n = len(f) - 1; if (n == -1) return {}; assert(f[0] == mint(0)); if (n == 0) return f; assert(f[1] != mint(0)); mint c = f[1]; mint ic = c.inverse(); for (auto& x: f) x *= ic; vc<mint> wt(n + 1); wt[n] = 1; vc<mint> A = power_projection<mint>(wt, f, n); vc<mint> g(n); FOR(i, 1, n + 1) g[n - i] = mint(n) * A[i] * inv<mint>(i); g = fps_pow_1<mint>(g, -inv<mint>(n)); g.insert(g.begin(), 0); mint pow = 1; FOR(i, len(g)) g[i] *= pow, pow *= ic; return g; } // G->F(G), G->DF(G) を与える // len(G) まで求める. len(F) まで求めてもいいよ. // 計算量は合成とだいたい同等 template <typename mint, typename F1, typename F2> vc<mint> compositional_inverse(const vc<mint>& F, F1 comp_F, F2 comp_DF) { const int N = len(F); assert(N <= 0 || F[0] == mint(0)); assert(N <= 1 || F[1] != mint(0)); vc<mint> G(2); G[1] = mint(1) / F[1]; while (len(G) < N) { int n = len(G); // G:= G(x)-(F(G(x))-x)/DF(G(x)) vc<mint> G2 = comp_DF(G); G.resize(2 * n); vc<mint> G1 = comp_F(G); G1 = {G1.begin() + n, G1.end()}; G1 = fps_div(G1, G2); FOR(i, n) G[n + i] -= G1[i]; } G.resize(N); return G; } #line 5 "graph/count/count_labeled_biconnected.hpp" // https://oeis.org/A013922 template <typename mint> vc<mint> count_labeled_biconnected(int N) { vc<mint> C = count_labeled_connected<mint>(N); FOR(i, N + 1) C[i] *= fact_inv<mint>(i); vc<mint> D(N); FOR(i, N) D[i] = C[i + 1] * mint(i + 1); vc<mint> E(N); FOR(i, N) E[i] = C[i] * mint(i); vc<mint> G = fps_log(D); vc<mint> IE = compositional_inverse(E); vc<mint> B = composition(G, IE); vc<mint> A = integrate(B); FOR(i, N + 1) A[i] *= fact<mint>(i); return A; } // https://oeis.org/A013922 template <typename mint> mint count_labeled_biconnected_single(int N) { if (N < 2) return 0; vc<mint> C = count_labeled_connected<mint>(N); FOR(i, N + 1) C[i] *= fact_inv<mint>(i); vc<mint> D(N); FOR(i, N) D[i] = C[i + 1] * mint(i + 1); vc<mint> E(N); FOR(i, N) E[i] = C[i] * mint(i); vc<mint> G = fps_log(D); // (N-1)[x^{N-1}]G(IE(x))=[x^{-1}]G'(x)E(x)^{-(N-1)} // =[x^{N-2}]G'(x)(E(x)/x)^{-(N-1)} G = differentiate(G); E.erase(E.begin()); E = fps_pow_1<mint>(E, -(N - 1)); mint ANS = 0; FOR(i, N - 1) ANS += G[i] * E[N - 2 - i]; ANS *= inv<mint>(N - 1); ANS *= inv<mint>(N); ANS *= fact<mint>(N); return ANS; } #line 7 "test/1_mytest/count_labeled_biconnected.test.cpp" using mint = modint998; void test() { vc<mint> F = count_labeled_biconnected<mint>(10); vi ANS = { 0, 0, 1, 1, 10, 238, 11368, 1014888, 166537616, 50680432112, 29107809374336, }; FOR(i, 11) assert(F[i] == mint(ANS[i])); FOR(i, 11) { mint a = count_labeled_biconnected_single<mint>(i); assert(a == mint(ANS[i])); } } void solve() { int a, b; cin >> a >> b; cout << a + b << '\n'; } signed main() { test(); solve(); return 0; }