//#define _GLIBCXX_DEBUG #include using namespace std; #define endl '\n' #define lfs cout<= (ll)(n); i--) using ll = long long; using ld = long double; const ll MOD1 = 1e9+7; const ll MOD9 = 998244353; const ll INF = 1e18; using P = pair; templatebool chmin(T1 &a,T2 b){if(a>b){a=b;return true;}else return false;} templatebool chmax(T1 &a,T2 b){if(avoid ans(bool x,T1 y,T2 z){if(x)cout<void debug(vector>&v,ll h,ll w){for(ll i=0;i&v,ll h,ll w){for(ll i=0;ivoid debug(vector&v,ll n){if(n!=0)cout<vector>vec(ll x, ll y, T w){vector>v(x,vector(y,w));return v;} ll gcd(ll x,ll y){ll r;while(y!=0&&(r=x%y)!=0){x=y;y=r;}return y==0?x:y;} vectordx={1,-1,0,0,1,1,-1,-1};vectordy={0,0,1,-1,1,-1,1,-1}; templatevector make_v(size_t a,T b){return vector(a,b);} templateauto make_v(size_t a,Ts... ts){return vector(a,make_v(ts...));} templateostream &operator<<(ostream &os, const pair&p){return os << p.first << " " << p.second;} templateostream &operator<<(ostream &os, const vector &v){for(auto &z:v)os << z << " ";cout<<"|"; return os;} templatevoid rearrange(vector&ord, vector&v){ auto tmp = v; for(int i=0;ivoid rearrange(vector&ord,Head&& head, Tail&&... tail){ rearrange(ord, head); rearrange(ord, tail...); } //mt19937 mt(chrono::steady_clock::now().time_since_epoch().count()); int popcount(ll x){return __builtin_popcountll(x);}; int poplow(ll x){return __builtin_ctzll(x);}; int pophigh(ll x){return 63 - __builtin_clzll(x);}; template< int mod > struct ModInt { int x; ModInt() : x(0) {} ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {} ModInt &operator+=(const ModInt &p) { if((x += p.x) >= mod) x -= mod; return *this; } ModInt &operator-=(const ModInt &p) { if((x += mod - p.x) >= mod) x -= mod; return *this; } ModInt &operator*=(const ModInt &p) { x = (int) (1LL * x * p.x % mod); return *this; } ModInt &operator/=(const ModInt &p) { *this *= p.inverse(); return *this; } ModInt operator-() const { return ModInt(-x); } friend ModInt operator+(const ModInt& lhs, const ModInt& rhs) { return ModInt(lhs) += rhs; } friend ModInt operator-(const ModInt& lhs, const ModInt& rhs) { return ModInt(lhs) -= rhs; } friend ModInt operator*(const ModInt& lhs, const ModInt& rhs) { return ModInt(lhs) *= rhs; } friend ModInt operator/(const ModInt& lhs, const ModInt& rhs) { return ModInt(lhs) /= rhs; } bool operator==(const ModInt &p) const { return x == p.x; } bool operator!=(const ModInt &p) const { return x != p.x; } ModInt inverse() const { int a = x, 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(int64_t n) const { ModInt ret(1), mul(x); while(n > 0) { if(n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } friend ostream &operator<<(ostream &os, const ModInt &p) { return os << p.x; } friend istream &operator>>(istream &is, ModInt &a) { int64_t t; is >> t; a = ModInt< mod >(t); return (is); } static int get_mod() { return mod; } }; using modint = ModInt< MOD9 >;modint pow(ll n, ll x){return modint(n).pow(x);}modint pow(modint n, ll x){return n.pow(x);} //using modint=ld; template< typename T > struct FormalPowerSeries : vector< T > { using vector< T >::vector; using P = FormalPowerSeries; using MULT = function< P(P, P) >; using FFT = function< void(P &) >; using SQRT = function< T(T) >; static MULT &get_mult() { static MULT mult = nullptr; return mult; } static void set_mult(MULT f) { get_mult() = f; } static FFT &get_fft() { static FFT fft = nullptr; return fft; } static FFT &get_ifft() { static FFT ifft = nullptr; return ifft; } static void set_fft(FFT f, FFT g) { get_fft() = f; get_ifft() = g; } static SQRT &get_sqrt() { static SQRT sqr = nullptr; return sqr; } static void set_sqrt(SQRT sqr) { get_sqrt() = sqr; } 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 < r.size(); i++) (*this)[i] += r[i]; return *this; } P &operator+=(const T &r) { if(this->empty()) this->resize(1); (*this)[0] += r; return *this; } P &operator-=(const P &r) { if(r.size() > this->size()) this->resize(r.size()); for(int i = 0; i < r.size(); i++) (*this)[i] -= r[i]; //shrink(); return *this; } P &operator-=(const T &r) { if(this->empty()) this->resize(1); (*this)[0] -= r; //shrink(); return *this; } P &operator*=(const T &v) { const int n = (int) this->size(); for(int k = 0; k < n; k++) (*this)[k] *= v; return *this; } P &operator*=(const P &r) { if(this->empty() || r.empty()) { this->clear(); return *this; } if(min(this->size(), r.size()) < 5 || get_mult() == nullptr){ P ret(this->size() + r.size() - 1); for(int i = 0; i < this->size(); i++){ for(int j = 0; j < r.size(); j++){ ret[i + j] += (*this)[i] * r[j]; } } return *this = ret; } return *this = get_mult()(*this, r); } P &operator%=(const P &r) { return *this -= *this / r * r; } P operator-() const { P ret(this->size()); for(int i = 0; i < this->size(); i++) ret[i] = -(*this)[i]; return 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 dot(P r) const { P ret(min(this->size(), r.size())); for(int i = 0; i < ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } P pre(int sz) const { return P(begin(*this), begin(*this) + min((int) this->size(), sz)); } P operator>>(int sz) const { if(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; } P rev(int deg = -1) const { P ret(*this); if(deg != -1) ret.resize(deg, T(0)); reverse(begin(ret), end(ret)); return ret; } 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; } // 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; if(get_fft() != nullptr) { P ret(*this); ret.resize(deg, T(0)); return ret.inv_fast(); } P ret({T(1) / (*this)[0]}); for(int i = 1; i < deg; i <<= 1) { ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1); } return ret.pre(deg); } // F(0) must be 1 P log(int deg = -1) const { assert((*this)[0] == 1); const int n = (int) this->size(); if(deg == -1) deg = n; return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } P sqrt(int deg = -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); if(ret.empty()) return {}; ret = ret << (i / 2); if(ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return P(deg, 0); } P ret; if(get_sqrt() == nullptr) { assert((*this)[0] == T(1)); ret = {T(1)}; } else { auto sqr = get_sqrt()((*this)[0]); if(sqr * sqr != (*this)[0]) return {}; ret = {T(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); } // F(0) must be 0 P exp(int deg = -1) const { assert((*this)[0] == T(0)); const int n = (int) this->size(); if(deg == -1) deg = n; if(get_fft() != nullptr) { P ret(*this); ret.resize(deg, T(0)); return ret.exp_rec(); } P ret({T(1)}); for(int i = 1; i < deg; i <<= 1) { ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1); } return ret.pre(deg); } P online_convolution_exp(const P &conv_coeff) const { const int n = (int) conv_coeff.size(); assert((n & (n - 1)) == 0); vector< P > conv_ntt_coeff; auto& fft = get_fft(); auto& ifft = get_ifft(); for(int i = n; i >= 1; i >>= 1) { P g(conv_coeff.pre(i)); fft(g); conv_ntt_coeff.emplace_back(g); } P conv_arg(n), conv_ret(n); auto rec = [&](auto rec, int l, int r, int d) -> void { if(r - l <= 16) { for(int i = l; i < r; i++) { T sum = 0; for(int j = l; j < i; j++) sum += conv_arg[j] * conv_coeff[i - j]; conv_ret[i] += sum; conv_arg[i] = i == 0 ? T(1) : conv_ret[i] / i; } } else { int m = (l + r) / 2; rec(rec, l, m, d + 1); P pre(r - l); for(int i = 0; i < m - l; i++) pre[i] = conv_arg[l + i]; fft(pre); for(int i = 0; i < r - l; i++) pre[i] *= conv_ntt_coeff[d][i]; ifft(pre); for(int i = 0; i < r - m; i++) conv_ret[m + i] += pre[m + i - l]; rec(rec, m, r, d + 1); } }; rec(rec, 0, n, 0); return conv_arg; } P exp_rec() const { assert((*this)[0] == T(0)); const int n = (int) this->size(); int m = 1; while(m < n) m *= 2; P conv_coeff(m); for(int i = 1; i < n; i++) conv_coeff[i] = (*this)[i] * i; return online_convolution_exp(conv_coeff).pre(n); } P inv_fast() const { assert(((*this)[0]) != T(0)); const int n = (int) this->size(); P res{T(1) / (*this)[0]}; for(int d = 1; d < n; 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]; get_fft()(f); get_fft()(g); for(int j = 0; j < 2 * d; j++) f[j] *= g[j]; get_ifft()(f); for(int j = 0; j < d; j++) { f[j] = 0; f[j + d] = -f[j + d]; } get_fft()(f); for(int j = 0; j < 2 * d; j++) f[j] *= g[j]; get_ifft()(f); for(int j = 0; j < d; j++) f[j] = res[j]; res = f; } return res.pre(n); } 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(deg) * k).exp(deg) * ((*this)[i].pow(k)); if(i * k > deg) return P(deg, T(0)); ret = (ret << (i * k)).pre(deg); if(ret.size() < deg) ret.resize(deg, T(0)); return ret; } } return *this; } T eval(T x) const { T r = 0, w = 1; for(auto &v : *this) { r += w * v; w *= x; } return r; } P pow_mod(int64_t n, P mod) const { P modinv = mod.rev().inv(); auto get_div = [&](P base) { if(base.size() < mod.size()) { base.clear(); return base; } int n = base.size() - mod.size() + 1; return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n); }; P x(*this), ret{1}; while(n > 0) { if(n & 1) { ret *= x; ret -= get_div(ret) * mod; } x *= x; x -= get_div(x) * mod; n >>= 1; } return ret; } void mul(vector> g, bool extend = false){ if(extend)this->resize(this->size() + g.back().first); int n = this->size(); int d = g[0].first; T c = g[0].second; if(d == 0)g.erase(g.begin()); else c = 0; for(int i = n - 1; i >= 0; i--){ (*this)[i] *= c; for(auto z : g){ if(z.first > i)continue; (*this)[i] += (*this)[i-z.first] * z.second; } } } void div(vector>g){//定数項は非ゼロ int n = this->size(); int d = g[0].first; T c = g[0].second; c = T(1) / c; g.erase(g.begin()); for(int i = 0; i < n; i++){ for(auto z : g){ if(z.first > i)continue; (*this)[i] -= (*this)[i-z.first] * z.second; } (*this)[i] *= c; } } }; template< typename T > FormalPowerSeries< T > stirling_second_kth_column(int N, int K) { FormalPowerSeries< T > poly(N + 1), ret(N + 1); poly[1] = 1; poly = poly.exp(); poly[0] -= 1; poly = poly.pow(K); T rev = 1, mul = 1; for(int i = 2; i <= K; i++) rev *= i; rev = T(1) / rev; poly *= rev; for(int i = 0; i <= N; i++) { ret[i] = poly[i] * mul; mul *= i + 1; } return ret; } template< typename Mint > struct NumberTheoreticTransformFriendlyModInt { vector< Mint > dw, idw; int max_base; Mint root; NumberTheoreticTransformFriendlyModInt() { 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++; root = 2; while(root.pow((mod - 1) >> 1) == 1) root += 1; assert(root.pow(mod - 1) == 1); dw.resize(max_base); idw.resize(max_base); for(int i = 0; i < max_base; i++) { dw[i] = -root.pow((mod - 1) >> (i + 2)); idw[i] = Mint(1) / dw[i]; } } void ntt(vector< Mint > &a) { const int n = (int) a.size(); assert((n & (n - 1)) == 0); assert(__builtin_ctz(n) <= max_base); for(int m = n; m >>= 1;) { Mint w = 1; for(int s = 0, k = 0; s < n; s += 2 * m) { for(int i = s, j = s + m; i < s + m; ++i, ++j) { auto x = a[i], y = a[j] * w; a[i] = x + y, a[j] = x - y; } w *= dw[__builtin_ctz(++k)]; } } } void intt(vector< Mint > &a, bool f = true) { const int n = (int) a.size(); assert((n & (n - 1)) == 0); assert(__builtin_ctz(n) <= max_base); for(int m = 1; m < n; m *= 2) { Mint w = 1; for(int s = 0, k = 0; s < n; s += 2 * m) { for(int i = s, j = s + m; i < s + m; ++i, ++j) { auto x = a[i], y = a[j]; a[i] = x + y, a[j] = (x - y) * w; } w *= idw[__builtin_ctz(++k)]; } } if(f) { Mint inv_sz = Mint(1) / n; for(int i = 0; i < n; i++) a[i] *= inv_sz; } } 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 T > struct Combination { vector< T > _fact, _rfact, _inv; Combination(ll sz) : _fact(sz + 1), _rfact(sz + 1), _inv(sz + 1) { _fact[0] = _rfact[sz] = _inv[0] = 1; for(ll i = 1; i <= sz; i++) _fact[i] = _fact[i - 1] * i; _rfact[sz] /= _fact[sz]; for(ll i = sz - 1; i >= 0; i--) _rfact[i] = _rfact[i + 1] * (i + 1); for(ll i = 1; i <= sz; i++) _inv[i] = _rfact[i] * _fact[i - 1]; } inline T fact(ll k) const { return _fact[k]; } inline T rfact(ll k) const { return _rfact[k]; } inline T inv(ll k) const { return _inv[k]; } T P(ll n, ll r) const { if(r < 0 || n < r) return 0; return fact(n) * rfact(n - r); } T C(ll p, ll q) const { if(q < 0 || p < q) return 0; return fact(p) * rfact(q) * rfact(p - q); } T RC(ll p, ll q) const { if(q < 0 || p < q) return 0; return rfact(p) * fact(q) * fact(p - q); } T H(ll n, ll r) const { if(n < 0 || r < 0) return (0); return r == 0 ? 1 : C(n + r - 1, r); } }; using Comb=Combination; int main(){ cin.tie(nullptr); ios_base::sync_with_stdio(false); ll res=0,buf=0; bool judge = true; using FPS=FormalPowerSeries; NumberTheoreticTransformFriendlyModInt ntt; auto mult=[&](const FPS &x,const FPS &y){ auto ret = ntt.multiply(x,y); return FPS(ret.begin(),ret.end()); }; FPS::set_mult(mult); FPS::set_fft([&](FPS &a){return ntt.ntt(a);},[&](FPS &b){return ntt.intt(b);}); ll n,m,k;cin>>n>>m>>k; FPS str=stirling_second_kth_column(n+1,k); Comb comb(n+2); modint ret=0; rep(i,1,n+1){ //cout<