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:heavy_check_mark: graph/count/count_labeled_bridgeless.hpp

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Code

#include "graph/count/count_labeled_connected.hpp"
#include "poly/compositional_inverse.hpp"
#include "poly/fps_exp.hpp"
#include "poly/fps_pow.hpp"

// 橋のない連結グラフ
// https://oeis.org/A095983
// N=1: 1
// O(Nlog^2N)
template <typename mint>
vc<mint> count_labeled_bridgeless(int N) {
  vc<mint> C = count_labeled_connected<mint>(N);
  FOR(i, N + 1) C[i] *= fact_inv<mint>(i);

  vc<mint> D(N + 1);
  FOR(i, N + 1) D[i] = mint(i) * C[i];

  vc<mint> E = fps_exp(D);
  E.insert(E.begin(), mint(0));
  E.pop_back();

  // D(x)=B(E(x))
  vc<mint> IE = compositional_inverse(E);
  vc<mint> B = composition(D, IE);

  vc<mint> A(N + 1);
  FOR(i, 1, N + 1) A[i] = B[i] * inv<mint>(i);

  FOR(i, 1, N + 1) A[i] *= fact<mint>(i);
  return A;
}

// https://oeis.org/A095983
// N での値のみ, O(NlogN)
template <typename mint>
mint count_labeled_bridgeless_single(int N) {
  if (N == 0) return 0;
  vc<mint> C = count_labeled_connected<mint>(N);
  FOR(i, N + 1) C[i] *= fact_inv<mint>(i);

  vc<mint> D(N + 1);
  FOR(i, N + 1) D[i] = mint(i) * C[i];

  vc<mint> E = fps_exp(D);
  E.insert(E.begin(), mint(0));
  E.pop_back();

  // D(x)=B(E(x))
  // [x^N]B(x) を求めたい
  // Lagrange Inversion
  // N[x^N]D(IE(x))=[x^{-1}]D'(x)E(x)^{-N}
  // =[x^{N-1}]D'(x)(E(x)/x)^{-N}

  E.erase(E.begin());
  E = fps_pow_1<mint>(E, -N);
  D = differentiate(D);
  mint ANS = 0;
  FOR(i, N) ANS += D[i] * E[N - 1 - i];
  ANS *= inv<mint>(N);

  // [x^N]B(x) が出た
  ANS *= inv<mint>(N);
  ANS *= fact<mint>(N);
  return ANS;
}
#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/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 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 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 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_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_bridgeless.hpp"

// 橋のない連結グラフ
// https://oeis.org/A095983
// N=1: 1
// O(Nlog^2N)
template <typename mint>
vc<mint> count_labeled_bridgeless(int N) {
  vc<mint> C = count_labeled_connected<mint>(N);
  FOR(i, N + 1) C[i] *= fact_inv<mint>(i);

  vc<mint> D(N + 1);
  FOR(i, N + 1) D[i] = mint(i) * C[i];

  vc<mint> E = fps_exp(D);
  E.insert(E.begin(), mint(0));
  E.pop_back();

  // D(x)=B(E(x))
  vc<mint> IE = compositional_inverse(E);
  vc<mint> B = composition(D, IE);

  vc<mint> A(N + 1);
  FOR(i, 1, N + 1) A[i] = B[i] * inv<mint>(i);

  FOR(i, 1, N + 1) A[i] *= fact<mint>(i);
  return A;
}

// https://oeis.org/A095983
// N での値のみ, O(NlogN)
template <typename mint>
mint count_labeled_bridgeless_single(int N) {
  if (N == 0) return 0;
  vc<mint> C = count_labeled_connected<mint>(N);
  FOR(i, N + 1) C[i] *= fact_inv<mint>(i);

  vc<mint> D(N + 1);
  FOR(i, N + 1) D[i] = mint(i) * C[i];

  vc<mint> E = fps_exp(D);
  E.insert(E.begin(), mint(0));
  E.pop_back();

  // D(x)=B(E(x))
  // [x^N]B(x) を求めたい
  // Lagrange Inversion
  // N[x^N]D(IE(x))=[x^{-1}]D'(x)E(x)^{-N}
  // =[x^{N-1}]D'(x)(E(x)/x)^{-N}

  E.erase(E.begin());
  E = fps_pow_1<mint>(E, -N);
  D = differentiate(D);
  mint ANS = 0;
  FOR(i, N) ANS += D[i] * E[N - 1 - i];
  ANS *= inv<mint>(N);

  // [x^N]B(x) が出た
  ANS *= inv<mint>(N);
  ANS *= fact<mint>(N);
  return ANS;
}
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