形式的冪級数の練習！
問題概要
各 に対して、次の問に答えよ。
二分木 (完全二分木でなくてもよいし、頂点数も未定) であって、
- 各頂点の重みが
のいずれか
- 各頂点の重みの総和が
であるようなものの個数を 998244353 で割ったあまりを求めよ。下図は 、
の場合を表している (9 通り)。
制約
考えたこと
これはもう解答見ながらやった。答えを とすると、
(
= {
と表せる。これを形式的冪級数の言葉で解釈してみよう。
とする。ここで、上の漸化式で の場合を特別扱いする必要があることに注意しよう。というのも、
の式に
を代入すると
となってしまう。しかし実際は
なので、
の場合を特別扱いする必要がありそう。まず
のときは、
と表せる。さらに定数項 () を合わせると、最終的に次にようになる。
これは についての二次方程式になっているので、解くと
となる。このままでは のとき分母が 0 になって不味いので式変形すると、次のようになる。
ここで、 を満たす方のみを取り出すと、こうなる。
あとは、形式的冪級数の sqrt 演算、逆元計算などを駆使して求めることができる。計算量は となる。
#include <bits/stdc++.h> using namespace std; // modint template<int MOD> struct Fp { long long val; constexpr Fp(long long v = 0) noexcept : val(v % MOD) { if (val < 0) val += MOD; } constexpr int getmod() const { return MOD; } constexpr Fp operator - () const noexcept { return val ? MOD - val : 0; } constexpr Fp operator + (const Fp& r) const noexcept { return Fp(*this) += r; } constexpr Fp operator - (const Fp& r) const noexcept { return Fp(*this) -= r; } constexpr Fp operator * (const Fp& r) const noexcept { return Fp(*this) *= r; } constexpr Fp operator / (const Fp& r) const noexcept { return Fp(*this) /= r; } constexpr Fp& operator += (const Fp& r) noexcept { val += r.val; if (val >= MOD) val -= MOD; return *this; } constexpr Fp& operator -= (const Fp& r) noexcept { val -= r.val; if (val < 0) val += MOD; return *this; } constexpr Fp& operator *= (const Fp& r) noexcept { val = val * r.val % MOD; return *this; } constexpr Fp& operator /= (const Fp& r) noexcept { long long a = r.val, b = MOD, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b, swap(a, b); u -= t * v, swap(u, v); } val = val * u % MOD; if (val < 0) val += MOD; return *this; } constexpr bool operator == (const Fp& r) const noexcept { return this->val == r.val; } constexpr bool operator != (const Fp& r) const noexcept { return this->val != r.val; } friend constexpr istream& operator >> (istream& is, Fp<MOD>& x) noexcept { is >> x.val; x.val %= MOD; if (x.val < 0) x.val += MOD; return is; } friend constexpr ostream& operator << (ostream& os, const Fp<MOD>& x) noexcept { return os << x.val; } friend constexpr Fp<MOD> modpow(const Fp<MOD>& r, long long n) noexcept { if (n == 0) return 1; auto t = modpow(r, n / 2); t = t * t; if (n & 1) t = t * r; return t; } friend constexpr Fp<MOD> modinv(const Fp<MOD>& r) noexcept { long long a = r.val, b = MOD, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b, swap(a, b); u -= t * v, swap(u, v); } return Fp<MOD>(u); } }; namespace NTT { long long modpow(long long a, long long n, int mod) { long long res = 1; while (n > 0) { if (n & 1) res = res * a % mod; a = a * a % mod; n >>= 1; } return res; } long long modinv(long long a, int mod) { long long b = mod, u = 1, v = 0; while (b) { long long t = a / b; a -= t * b, swap(a, b); u -= t * v, swap(u, v); } u %= mod; if (u < 0) u += mod; return u; } int calc_primitive_root(int mod) { if (mod == 2) return 1; if (mod == 167772161) return 3; if (mod == 469762049) return 3; if (mod == 754974721) return 11; if (mod == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; long long x = (mod - 1) / 2; while (x % 2 == 0) x /= 2; for (long long i = 3; i * i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) x /= i; } } if (x > 1) divs[cnt++] = x; for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (modpow(g, (mod - 1) / divs[i], mod) == 1) { ok = false; break; } } if (ok) return g; } } int get_fft_size(int N, int M) { int size_a = 1, size_b = 1; while (size_a < N) size_a <<= 1; while (size_b < M) size_b <<= 1; return max(size_a, size_b) << 1; } // number-theoretic transform template<class mint> void trans(vector<mint>& v, bool inv = false) { if (v.empty()) return; int N = (int)v.size(); int MOD = v[0].getmod(); int PR = calc_primitive_root(MOD); static bool first = true; static vector<long long> vbw(30), vibw(30); if (first) { first = false; for (int k = 0; k < 30; ++k) { vbw[k] = modpow(PR, (MOD - 1) >> (k + 1), MOD); vibw[k] = modinv(vbw[k], MOD); } } for (int i = 0, j = 1; j < N - 1; j++) { for (int k = N >> 1; k > (i ^= k); k >>= 1); if (i > j) swap(v[i], v[j]); } for (int k = 0, t = 2; t <= N; ++k, t <<= 1) { long long bw = vbw[k]; if (inv) bw = vibw[k]; for (int i = 0; i < N; i += t) { mint w = 1; for (int j = 0; j < t/2; ++j) { int j1 = i + j, j2 = i + j + t/2; mint c1 = v[j1], c2 = v[j2] * w; v[j1] = c1 + c2; v[j2] = c1 - c2; w *= bw; } } } if (inv) { long long invN = modinv(N, MOD); for (int i = 0; i < N; ++i) v[i] = v[i] * invN; } } // for garner static constexpr int MOD0 = 754974721; static constexpr int MOD1 = 167772161; static constexpr int MOD2 = 469762049; using mint0 = Fp<MOD0>; using mint1 = Fp<MOD1>; using mint2 = Fp<MOD2>; static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1); static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2); static const mint2 imod01 = 187290749; // imod1 / MOD0; // small case (T = mint, long long) template<class T> vector<T> naive_mul (const vector<T>& A, const vector<T>& B) { if (A.empty() || B.empty()) return {}; int N = (int)A.size(), M = (int)B.size(); vector<T> res(N + M - 1); for (int i = 0; i < N; ++i) for (int j = 0; j < M; ++j) res[i + j] += A[i] * B[j]; return res; } // mint template<class mint> vector<mint> mul (const vector<mint>& A, const vector<mint>& B) { if (A.empty() || B.empty()) return {}; int N = (int)A.size(), M = (int)B.size(); if (min(N, M) < 30) return naive_mul(A, B); int MOD = A[0].getmod(); int size_fft = get_fft_size(N, M); if (MOD == 998244353) { vector<mint> a(size_fft), b(size_fft), c(size_fft); for (int i = 0; i < N; ++i) a[i] = A[i]; for (int i = 0; i < M; ++i) b[i] = B[i]; trans(a), trans(b); vector<mint> res(size_fft); for (int i = 0; i < size_fft; ++i) res[i] = a[i] * b[i]; trans(res, true); res.resize(N + M - 1); return res; } vector<mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0); vector<mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0); vector<mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0); for (int i = 0; i < N; ++i) a0[i] = A[i].val, a1[i] = A[i].val, a2[i] = A[i].val; for (int i = 0; i < M; ++i) b0[i] = B[i].val, b1[i] = B[i].val, b2[i] = B[i].val; trans(a0), trans(a1), trans(a2), trans(b0), trans(b1), trans(b2); for (int i = 0; i < size_fft; ++i) { c0[i] = a0[i] * b0[i]; c1[i] = a1[i] * b1[i]; c2[i] = a2[i] * b2[i]; } trans(c0, true), trans(c1, true), trans(c2, true); static const mint mod0 = MOD0, mod01 = mod0 * MOD1; vector<mint> res(N + M - 1); for (int i = 0; i < N + M - 1; ++i) { int y0 = c0[i].val; int y1 = (imod0 * (c1[i] - y0)).val; int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val; res[i] = mod01 * y2 + mod0 * y1 + y0; } return res; } // long long vector<long long> mul_ll (const vector<long long>& A, const vector<long long>& B) { if (A.empty() || B.empty()) return {}; int N = (int)A.size(), M = (int)B.size(); if (min(N, M) < 30) return naive_mul(A, B); int size_fft = get_fft_size(N, M); vector<mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0); vector<mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0); vector<mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0); for (int i = 0; i < N; ++i) a0[i] = A[i], a1[i] = A[i], a2[i] = A[i]; for (int i = 0; i < M; ++i) b0[i] = B[i], b1[i] = B[i], b2[i] = B[i]; trans(a0), trans(a1), trans(a2), trans(b0), trans(b1), trans(b2); for (int i = 0; i < size_fft; ++i) { c0[i] = a0[i] * b0[i]; c1[i] = a1[i] * b1[i]; c2[i] = a2[i] * b2[i]; } trans(c0, true), trans(c1, true), trans(c2, true); static const long long mod0 = MOD0, mod01 = mod0 * MOD1; vector<long long> res(N + M - 1); for (int i = 0; i < N + M - 1; ++i) { int y0 = c0[i].val; int y1 = (imod0 * (c1[i] - y0)).val; int y2 = (imod01 * (c2[i] - y0) - imod1 * y1).val; res[i] = mod01 * y2 + mod0 * y1 + y0; } return res; } }; // Binomial Coefficient template<class T> struct BiCoef { vector<T> fact_, inv_, finv_; constexpr BiCoef() {} constexpr BiCoef(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) { init(n); } constexpr void init(int n) noexcept { fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1); int MOD = fact_[0].getmod(); for(int i = 2; i < n; i++){ fact_[i] = fact_[i-1] * i; inv_[i] = -inv_[MOD%i] * (MOD/i); finv_[i] = finv_[i-1] * inv_[i]; } } constexpr T com(int n, int k) const noexcept { if (n < k || n < 0 || k < 0) return 0; return fact_[n] * finv_[k] * finv_[n-k]; } constexpr T fact(int n) const noexcept { if (n < 0) return 0; return fact_[n]; } constexpr T inv(int n) const noexcept { if (n < 0) return 0; return inv_[n]; } constexpr T finv(int n) const noexcept { if (n < 0) return 0; return finv_[n]; } }; // Formal Power Series template <typename mint> struct FPS : vector<mint> { using vector<mint>::vector; // constructor FPS(const vector<mint>& r) : vector<mint>(r) {} // core operator inline FPS pre(int siz) const { return FPS(begin(*this), begin(*this) + min((int)this->size(), siz)); } inline FPS rev() const { FPS res = *this; reverse(begin(res), end(res)); return res; } inline FPS& normalize() { while (!this->empty() && this->back() == 0) this->pop_back(); return *this; } // basic operator inline FPS operator - () const noexcept { FPS res = (*this); for (int i = 0; i < (int)res.size(); ++i) res[i] = -res[i]; return res; } inline FPS operator + (const mint& v) const { return FPS(*this) += v; } inline FPS operator + (const FPS& r) const { return FPS(*this) += r; } inline FPS operator - (const mint& v) const { return FPS(*this) -= v; } inline FPS operator - (const FPS& r) const { return FPS(*this) -= r; } inline FPS operator * (const mint& v) const { return FPS(*this) *= v; } inline FPS operator * (const FPS& r) const { return FPS(*this) *= r; } inline FPS operator / (const mint& v) const { return FPS(*this) /= v; } inline FPS operator << (int x) const { return FPS(*this) <<= x; } inline FPS operator >> (int x) const { return FPS(*this) >>= x; } inline FPS& operator += (const mint& v) { if (this->empty()) this->resize(1); (*this)[0] += v; return *this; } inline FPS& operator += (const FPS& r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); ++i) (*this)[i] += r[i]; return *this; } inline FPS& operator -= (const mint& v) { if (this->empty()) this->resize(1); (*this)[0] -= v; return *this; } inline FPS& operator -= (const FPS& r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); ++i) (*this)[i] -= r[i]; return *this; } inline FPS& operator *= (const mint& v) { for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= v; return *this; } inline FPS& operator *= (const FPS& r) { return *this = NTT::mul((*this), r); } inline FPS& operator /= (const mint& v) { assert(v != 0); mint iv = modinv(v); for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= iv; return *this; } inline FPS& operator <<= (int x) { FPS res(x, 0); res.insert(res.end(), begin(*this), end(*this)); return *this = res; } inline FPS& operator >>= (int x) { FPS res; res.insert(res.end(), begin(*this) + x, end(*this)); return *this = res; } inline mint eval(const mint& v){ mint res = 0; for (int i = (int)this->size()-1; i >= 0; --i) { res *= v; res += (*this)[i]; } return res; } inline friend FPS gcd(const FPS& f, const FPS& g) { if (g.empty()) return f; return gcd(g, f % g); } // advanced operation // df/dx inline friend FPS diff(const FPS& f) { int n = (int)f.size(); FPS res(n-1); for (int i = 1; i < n; ++i) res[i-1] = f[i] * i; return res; } // \int f dx inline friend FPS integral(const FPS& f) { int n = (int)f.size(); FPS res(n+1, 0); for (int i = 0; i < n; ++i) res[i+1] = f[i] / (i+1); return res; } // inv(f), f[0] must not be 0 inline friend FPS inv(const FPS& f, int deg) { assert(f[0] != 0); if (deg < 0) deg = (int)f.size(); FPS res({mint(1) / f[0]}); for (int i = 1; i < deg; i <<= 1) { res = (res + res - res * res * f.pre(i << 1)).pre(i << 1); } res.resize(deg); return res; } inline friend FPS inv(const FPS& f) { return inv(f, f.size()); } // division, r must be normalized (r.back() must not be 0) inline FPS& operator /= (const FPS& r) { assert(!r.empty()); assert(r.back() != 0); this->normalize(); if (this->size() < r.size()) { this->clear(); return *this; } int need = (int)this->size() - (int)r.size() + 1; *this = ((*this).rev().pre(need) * inv(r.rev(), need)).pre(need).rev(); return *this; } inline FPS& operator %= (const FPS &r) { assert(!r.empty()); assert(r.back() != 0); this->normalize(); FPS q = (*this) / r; return *this -= q * r; } inline FPS operator / (const FPS& r) const { return FPS(*this) /= r; } inline FPS operator % (const FPS& r) const { return FPS(*this) %= r; } // log(f) = \int f'/f dx, f[0] must be 1 inline friend FPS log(const FPS& f, int deg) { assert(f[0] == 1); FPS res = integral(diff(f) * inv(f, deg)); res.resize(deg); return res; } inline friend FPS log(const FPS& f) { return log(f, f.size()); } // exp(f), f[0] must be 0 inline friend FPS exp(const FPS& f, int deg) { assert(f[0] == 0); FPS res(1, 1); for (int i = 1; i < deg; i <<= 1) { res = res * (f.pre(i<<1) - log(res, i<<1) + 1).pre(i<<1); } res.resize(deg); return res; } inline friend FPS exp(const FPS& f) { return exp(f, f.size()); } // pow(f) = exp(e * log f) inline friend FPS pow(const FPS& f, long long e, int deg) { long long i = 0; while (i < (int)f.size() && f[i] == 0) ++i; if (i == (int)f.size()) return FPS(deg, 0); if (i * e >= deg) return FPS(deg, 0); mint k = f[i]; FPS res = exp(log((f >> i) / k) * e) * modpow(k, e) << (e * i); res.resize(deg); return res; } inline friend FPS pow(const FPS& f, long long e) { return pow(f, e, f.size()); } // sqrt(f), f[0] must be 1 inline friend FPS sqrt_base(const FPS& f, int deg) { assert(f[0] == 1); mint inv2 = mint(1) / 2; FPS res(1, 1); for (int i = 1; i < deg; i <<= 1) { res = (res + f.pre(i << 1) * inv(res, i << 1)).pre(i << 1); for (mint& x : res) x *= inv2; } res.resize(deg); return res; } inline friend FPS sqrt_base(const FPS& f) { return sqrt_base(f, f.size()); } }; const int MOD = 998244353; using mint = Fp<MOD>; int main() { int N, M; cin >> N >> M; FPS<mint> C(M+1, 0); for (int i = 0; i < N; ++i) { int c; cin >> c; if (c > M) continue; C[c] += 1; } FPS<mint> F = inv(sqrt_base(C * mint(-4) + 1) + 1) * 2; for (int w = 1; w <= M; ++w) cout << F[w] << endl; }
