#include using namespace std; /** * @brief Montgomery ModInt */ template< uint32_t mod, bool fast = false > struct MontgomeryModInt { using mint = MontgomeryModInt; using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 ret = mod; for(i32 i = 0; i < 4; i++) ret *= 2 - mod * ret; return ret; } static constexpr u32 r = get_r(); static constexpr u32 n2 = -u64(mod) % mod; static_assert(r * mod == 1, "invalid, r * mod != 1"); static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30"); static_assert((mod & 1) == 1, "invalid, mod % 2 == 0"); u32 x; MontgomeryModInt() : x{} {} MontgomeryModInt(const i64 &a) : x(reduce(u64(fast ? a : (a % mod + mod)) * n2)) {} static constexpr u32 reduce(const u64 &b) { return u32(b >> 32) + mod - u32((u64(u32(b) * r) * mod) >> 32); } mint &operator+=(const mint &p) { if(i32(x += p.x - 2 * mod) < 0) x += 2 * mod; return *this; } mint &operator-=(const mint &p) { if(i32(x -= p.x) < 0) x += 2 * mod; return *this; } mint &operator*=(const mint &p) { x = reduce(u64(x) * p.x); return *this; } mint &operator/=(const mint &p) { *this *= p.inverse(); return *this; } mint operator-() const { return mint() - *this; } mint operator+(const mint &p) const { return mint(*this) += p; } mint operator-(const mint &p) const { return mint(*this) -= p; } mint operator*(const mint &p) const { return mint(*this) *= p; } mint operator/(const mint &p) const { return mint(*this) /= p; } bool operator==(const mint &p) const { return (x >= mod ? x - mod : x) == (p.x >= mod ? p.x - mod : p.x); } bool operator!=(const mint &p) const { return (x >= mod ? x - mod : x) != (p.x >= mod ? p.x - mod : p.x); } u32 get() const { u32 ret = reduce(x); return ret >= mod ? ret - mod : ret; } mint pow(u64 n) const { mint ret(1), mul(*this); while(n > 0) { if(n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } mint inverse() const { return pow(mod - 2); } friend ostream &operator<<(ostream &os, const mint &p) { return os << p.get(); } friend istream &operator>>(istream &is, mint &a) { i64 t; is >> t; a = mint(t); return is; } static u32 get_mod() { return mod; } }; using modint = MontgomeryModInt< 998244353 >; #line 1 "math/fft/number-theoretic-transform-friendly-mod-int.cpp" /** * @brief Number Theoretic Transform Friendly ModInt */ template< typename Mint > struct NumberTheoreticTransformFriendlyModInt { static vector< Mint > roots, iroots, rate3, irate3; static int max_base; NumberTheoreticTransformFriendlyModInt() = default; static void init() { if(roots.empty()) { const unsigned mod = Mint::get_mod(); assert(mod >= 3 && mod % 2 == 1); auto tmp = mod - 1; max_base = 0; while(tmp % 2 == 0) tmp >>= 1, max_base++; Mint root = 2; while(root.pow((mod - 1) >> 1) == 1) { root += 1; } assert(root.pow(mod - 1) == 1); roots.resize(max_base + 1); iroots.resize(max_base + 1); rate3.resize(max_base + 1); irate3.resize(max_base + 1); roots[max_base] = root.pow((mod - 1) >> max_base); iroots[max_base] = Mint(1) / roots[max_base]; for(int i = max_base - 1; i >= 0; i--) { roots[i] = roots[i + 1] * roots[i + 1]; iroots[i] = iroots[i + 1] * iroots[i + 1]; } { Mint prod = 1, iprod = 1; for(int i = 0; i <= max_base - 3; i++) { rate3[i] = roots[i + 3] * prod; irate3[i] = iroots[i + 3] * iprod; prod *= iroots[i + 3]; iprod *= roots[i + 3]; } } } } static void ntt(vector< Mint > &a) { init(); const int n = (int) a.size(); assert((n & (n - 1)) == 0); int h = __builtin_ctz(n); assert(h <= max_base); int len = 0; Mint imag = roots[2]; if(h & 1) { int p = 1 << (h - 1); Mint rot = 1; for(int i = 0; i < p; i++) { auto r = a[i + p]; a[i + p] = a[i] - r; a[i] += r; } len++; } for(; len + 1 < h; len += 2) { int p = 1 << (h - len - 2); { // s = 0 for(int i = 0; i < p; i++) { auto a0 = a[i]; auto a1 = a[i + p]; auto a2 = a[i + 2 * p]; auto a3 = a[i + 3 * p]; auto a1na3imag = (a1 - a3) * imag; auto a0a2 = a0 + a2; auto a1a3 = a1 + a3; auto a0na2 = a0 - a2; a[i] = a0a2 + a1a3; a[i + 1 * p] = a0a2 - a1a3; a[i + 2 * p] = a0na2 + a1na3imag; a[i + 3 * p] = a0na2 - a1na3imag; } } Mint rot = rate3[0]; for(int s = 1; s < (1 << len); s++) { int offset = s << (h - len); Mint rot2 = rot * rot; Mint rot3 = rot2 * rot; for(int i = 0; i < p; i++) { auto a0 = a[i + offset]; auto a1 = a[i + offset + p] * rot; auto a2 = a[i + offset + 2 * p] * rot2; auto a3 = a[i + offset + 3 * p] * rot3; auto a1na3imag = (a1 - a3) * imag; auto a0a2 = a0 + a2; auto a1a3 = a1 + a3; auto a0na2 = a0 - a2; a[i + offset] = a0a2 + a1a3; a[i + offset + 1 * p] = a0a2 - a1a3; a[i + offset + 2 * p] = a0na2 + a1na3imag; a[i + offset + 3 * p] = a0na2 - a1na3imag; } rot *= rate3[__builtin_ctz(~s)]; } } } static void intt(vector< Mint > &a, bool f = true) { init(); const int n = (int) a.size(); assert((n & (n - 1)) == 0); int h = __builtin_ctz(n); assert(h <= max_base); int len = h; Mint iimag = iroots[2]; for(; len > 1; len -= 2) { int p = 1 << (h - len); { // s = 0 for(int i = 0; i < p; i++) { auto a0 = a[i]; auto a1 = a[i + 1 * p]; auto a2 = a[i + 2 * p]; auto a3 = a[i + 3 * p]; auto a2na3iimag = (a2 - a3) * iimag; auto a0na1 = a0 - a1; auto a0a1 = a0 + a1; auto a2a3 = a2 + a3; a[i] = a0a1 + a2a3; a[i + 1 * p] = (a0na1 + a2na3iimag); a[i + 2 * p] = (a0a1 - a2a3); a[i + 3 * p] = (a0na1 - a2na3iimag); } } Mint irot = irate3[0]; for(int s = 1; s < (1 << (len - 2)); s++) { int offset = s << (h - len + 2); Mint irot2 = irot * irot; Mint irot3 = irot2 * irot; for(int i = 0; i < p; i++) { auto a0 = a[i + offset]; auto a1 = a[i + offset + 1 * p]; auto a2 = a[i + offset + 2 * p]; auto a3 = a[i + offset + 3 * p]; auto a2na3iimag = (a2 - a3) * iimag; auto a0na1 = a0 - a1; auto a0a1 = a0 + a1; auto a2a3 = a2 + a3; a[i + offset] = a0a1 + a2a3; a[i + offset + 1 * p] = (a0na1 + a2na3iimag) * irot; a[i + offset + 2 * p] = (a0a1 - a2a3) * irot2; a[i + offset + 3 * p] = (a0na1 - a2na3iimag) * irot3; } irot *= irate3[__builtin_ctz(~s)]; } } if(len >= 1) { int p = 1 << (h - 1); for(int i = 0; i < p; i++) { auto ajp = a[i] - a[i + p]; a[i] += a[i + p]; a[i + p] = ajp; } } if(f) { Mint inv_sz = Mint(1) / n; for(int i = 0; i < n; i++) a[i] *= inv_sz; } } static vector< Mint > multiply(vector< Mint > a, vector< Mint > b) { int need = a.size() + b.size() - 1; int nbase = 1; while((1 << nbase) < need) nbase++; int sz = 1 << nbase; a.resize(sz, 0); b.resize(sz, 0); ntt(a); ntt(b); Mint inv_sz = Mint(1) / sz; for(int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz; intt(a, false); a.resize(need); return a; } }; template< typename Mint > vector< Mint > NumberTheoreticTransformFriendlyModInt< Mint >::roots = vector< Mint >(); template< typename Mint > vector< Mint > NumberTheoreticTransformFriendlyModInt< Mint >::iroots = vector< Mint >(); template< typename Mint > vector< Mint > NumberTheoreticTransformFriendlyModInt< Mint >::rate3 = vector< Mint >(); template< typename Mint > vector< Mint > NumberTheoreticTransformFriendlyModInt< Mint >::irate3 = vector< Mint >(); template< typename Mint > int NumberTheoreticTransformFriendlyModInt< Mint >::max_base = 0; #line 2 "math/fps/formal-power-series-friendly-ntt.cpp" /** * @brief Formal Power Series Friendly NTT(NTTmod用形式的冪級数) * @docs docs/formal-power-series-friendly-ntt.md */ template< typename T > struct FormalPowerSeriesFriendlyNTT : vector< T > { using vector< T >::vector; using P = FormalPowerSeriesFriendlyNTT; using NTT = NumberTheoreticTransformFriendlyModInt< T >; P pre(int deg) const { return P(begin(*this), begin(*this) + min((int) this->size(), deg)); } P rev(int deg = -1) const { P ret(*this); if(deg != -1) ret.resize(deg, T(0)); reverse(begin(ret), end(ret)); return ret; } void shrink() { while(this->size() && this->back() == T(0)) this->pop_back(); } P operator+(const P &r) const { return P(*this) += r; } P operator+(const T &v) const { return P(*this) += v; } P operator-(const P &r) const { return P(*this) -= r; } P operator-(const T &v) const { return P(*this) -= v; } P operator*(const P &r) const { return P(*this) *= r; } P operator*(const T &v) const { return P(*this) *= v; } P operator/(const P &r) const { return P(*this) /= r; } P operator%(const P &r) const { return P(*this) %= r; } P &operator+=(const P &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; } P &operator-=(const P &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; } // https://judge.yosupo.jp/problem/convolution_mod P &operator*=(const P &r) { if(this->empty() || r.empty()) { this->clear(); return *this; } auto ret = NTT::multiply(*this, r); return *this = {begin(ret), end(ret)}; } P &operator/=(const P &r) { if(this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n); } P &operator%=(const P &r) { *this -= *this / r * r; shrink(); return *this; } // https://judge.yosupo.jp/problem/division_of_polynomials pair< P, P > div_mod(const P &r) { P q = *this / r; P x = *this - q * r; x.shrink(); return make_pair(q, x); } P operator-() const { P ret(this->size()); for(int i = 0; i < (int) this->size(); i++) ret[i] = -(*this)[i]; return ret; } P &operator+=(const T &r) { if(this->empty()) this->resize(1); (*this)[0] += r; return *this; } P &operator-=(const T &r) { if(this->empty()) this->resize(1); (*this)[0] -= r; return *this; } P &operator*=(const T &v) { for(int i = 0; i < (int) this->size(); i++) (*this)[i] *= v; return *this; } P dot(P r) const { P ret(min(this->size(), r.size())); for(int i = 0; i < (int) ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } P operator>>(int sz) const { if((int) this->size() <= sz) return {}; P ret(*this); ret.erase(ret.begin(), ret.begin() + sz); return ret; } P operator<<(int sz) const { P ret(*this); ret.insert(ret.begin(), sz, T(0)); return ret; } T operator()(T x) const { T r = 0, w = 1; for(auto &v : *this) { r += w * v; w *= x; } return r; } P diff() const { const int n = (int) this->size(); P ret(max(0, n - 1)); for(int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i); return ret; } P integral() const { const int n = (int) this->size(); P ret(n + 1); ret[0] = T(0); for(int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1); return ret; } // https://judge.yosupo.jp/problem/inv_of_formal_power_series // F(0) must not be 0 P inv(int deg = -1) const { assert(((*this)[0]) != T(0)); const int n = (int) this->size(); if(deg == -1) deg = n; P res(deg); res[0] = {T(1) / (*this)[0]}; for(int d = 1; d < deg; d <<= 1) { P f(2 * d), g(2 * d); for(int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j]; for(int j = 0; j < d; j++) g[j] = res[j]; NTT::ntt(f); NTT::ntt(g); f = f.dot(g); NTT::intt(f); for(int j = 0; j < d; j++) f[j] = 0; NTT::ntt(f); for(int j = 0; j < 2 * d; j++) f[j] *= g[j]; NTT::intt(f); for(int j = d; j < min(2 * d, deg); j++) res[j] = -f[j]; } return res; } // https://judge.yosupo.jp/problem/log_of_formal_power_series // F(0) must be 1 P log(int deg = -1) const { assert((*this)[0] == T(1)); const int n = (int) this->size(); if(deg == -1) deg = n; return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series P sqrt(int deg = -1, const function< T(T) > &get_sqrt = [](T) { return T(1); }) const { const int n = (int) this->size(); if(deg == -1) deg = n; if((*this)[0] == T(0)) { for(int i = 1; i < n; i++) { if((*this)[i] != T(0)) { if(i & 1) return {}; if(deg - i / 2 <= 0) break; auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt); if(ret.empty()) return {}; ret = ret << (i / 2); if((int) ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return P(deg, 0); } auto sqr = T(get_sqrt((*this)[0])); if(sqr * sqr != (*this)[0]) return {}; P ret{sqr}; T inv2 = T(1) / T(2); for(int i = 1; i < deg; i <<= 1) { ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2; } return ret.pre(deg); } P sqrt(const function< T(T) > &get_sqrt, int deg = -1) const { return sqrt(deg, get_sqrt); } // https://judge.yosupo.jp/problem/exp_of_formal_power_series // F(0) must be 0 P exp(int deg = -1) const { if(deg == -1) deg = this->size(); assert((*this)[0] == T(0)); P inv; inv.reserve(deg + 1); inv.push_back(T(0)); inv.push_back(T(1)); auto inplace_integral = [&](P &F) -> void { const int n = (int) F.size(); auto mod = T::get_mod(); while((int) inv.size() <= n) { int i = inv.size(); inv.push_back((-inv[mod % i]) * (mod / i)); } F.insert(begin(F), T(0)); for(int i = 1; i <= n; i++) F[i] *= inv[i]; }; auto inplace_diff = [](P &F) -> void { if(F.empty()) return; F.erase(begin(F)); T coeff = 1, one = 1; for(int i = 0; i < (int) F.size(); i++) { F[i] *= coeff; coeff += one; } }; P b{1, 1 < (int) this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1}; for(int m = 2; m < deg; m *= 2) { auto y = b; y.resize(2 * m); NTT::ntt(y); z1 = z2; P z(m); for(int i = 0; i < m; ++i) z[i] = y[i] * z1[i]; NTT::intt(z); fill(begin(z), begin(z) + m / 2, T(0)); NTT::ntt(z); for(int i = 0; i < m; ++i) z[i] *= -z1[i]; NTT::intt(z); c.insert(end(c), begin(z) + m / 2, end(z)); z2 = c; z2.resize(2 * m); NTT::ntt(z2); P x(begin(*this), begin(*this) + min< int >(this->size(), m)); inplace_diff(x); x.push_back(T(0)); NTT::ntt(x); for(int i = 0; i < m; ++i) x[i] *= y[i]; NTT::intt(x); x -= b.diff(); x.resize(2 * m); for(int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0); NTT::ntt(x); for(int i = 0; i < 2 * m; ++i) x[i] *= z2[i]; NTT::intt(x); x.pop_back(); inplace_integral(x); for(int i = m; i < min< int >(this->size(), 2 * m); ++i) x[i] += (*this)[i]; fill(begin(x), begin(x) + m, T(0)); NTT::ntt(x); for(int i = 0; i < 2 * m; ++i) x[i] *= y[i]; NTT::intt(x); b.insert(end(b), begin(x) + m, end(x)); } return P{begin(b), begin(b) + deg}; } // https://judge.yosupo.jp/problem/pow_of_formal_power_series P pow(int64_t k, int deg = -1) const { const int n = (int) this->size(); if(deg == -1) deg = n; for(int i = 0; i < n; i++) { if((*this)[i] != T(0)) { T rev = T(1) / (*this)[i]; P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k)); if(i * k > deg) return P(deg, T(0)); ret = (ret << (i * k)).pre(deg); if((int) ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return *this; } P mod_pow(int64_t k, P g) const { P modinv = g.rev().inv(); auto get_div = [&](P base) { if(base.size() < g.size()) { base.clear(); return base; } int n = base.size() - g.size() + 1; return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n); }; P x(*this), ret{1}; while(k > 0) { if(k & 1) { ret *= x; ret -= get_div(ret) * g; ret.shrink(); } x *= x; x -= get_div(x) * g; x.shrink(); k >>= 1; } return ret; } // https://judge.yosupo.jp/problem/polynomial_taylor_shift P taylor_shift(T c) const { int n = (int) this->size(); vector< T > fact(n), rfact(n); fact[0] = rfact[0] = T(1); for(int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i); rfact[n - 1] = T(1) / fact[n - 1]; for(int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i); P p(*this); for(int i = 0; i < n; i++) p[i] *= fact[i]; p = p.rev(); P bs(n, T(1)); for(int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1]; p = (p * bs).pre(n); p = p.rev(); for(int i = 0; i < n; i++) p[i] *= rfact[i]; return p; } }; template< template< typename > class FPS, typename Mint > Mint coeff_of_rational_function(FPS< Mint > P, FPS< Mint > Q, int64_t k) { // compute the coefficient [x^k] P/Q of rational power series Mint ret = 0; if(P.size() >= Q.size()) { auto R = P / Q; P -= R * Q; P.shrink(); if(k < (int) R.size()) ret += R[k]; } if(P.empty()) return ret; P.resize((int) Q.size() - 1); auto sub = [&](const FPS< Mint > &as, bool odd) { FPS< Mint > bs((as.size() + !odd) / 2); for(int i = odd; i < (int) as.size(); i += 2) bs[i >> 1] = as[i]; return bs; }; while(k > 0) { auto Q2(Q); for(int i = 1; i < (int) Q2.size(); i += 2) Q2[i] = -Q2[i]; P = sub(P * Q2, k & 1); Q = sub(Q * Q2, 0); k >>= 1; } return ret + P[0]; } int main(){ int N, M; cin >> N >> M; assert(1 <= N && N <= (int)1e9); assert(1 <= M && M <= 100); using FPS = FormalPowerSeriesFriendlyNTT; FPS denominator = {1}; vectorf(M + 1); for(int i = 1; i <= M; i++){ f[i] = FPS(i + 1, 0); f[i][0] = 1; f[i][i] = -1; } vectorff(M + 1); for(int i = 2; i <= M; i++){ ff[i] = f[1] * f[i] + f[i] - f[1]; denominator *= ff[i]; } FPS numerator = denominator; for(int i = 2; i <= M; i++){ FPS add = {1}; for(int j = 2; j <= M; j++){ if(i != j){ add *= ff[j]; } } numerator -= add * f[i]; numerator += add * f[1]; } cout << coeff_of_rational_function(denominator, numerator, N) <