#include //#include //#pragma GCC optimize("O3") using namespace std; #define reps(i,s,n) for(int i = s; i < n; i++) #define rep(i,n) reps(i,0,n) #define Rreps(i,n,e) for(int i = n - 1; i >= e; --i) #define Rrep(i,n) Rreps(i,n,0) #define ALL(a) a.begin(), a.end() using ll = long long; using vec = vector; using mat = vector; ll N,M,H,W,Q,K,A,B; string S; using P = pair; const ll INF = (1LL<<60); template bool chmin(T &a, const T &b){ if(a > b) {a = b; return true;} else return false; } template bool chmax(T &a, const T &b){ if(a < b) {a = b; return true;} else return false; } template void my_printv(std::vector v,bool endline = true){ if(!v.empty()){ for(std::size_t i{}; i; template class modint{ public: ll x; constexpr modint(){x = 0;} constexpr modint(ll _x) : x((_x < 0 ? ((_x += (LLONG_MAX / mod) * mod) < 0 ? _x + (LLONG_MAX / mod) * mod : _x) : _x)%mod){} constexpr modint set_raw(ll _x){ //_x in [0, mod) x = _x; return *this; } constexpr modint operator-(){ return x == 0 ? 0 : mod - x; } constexpr modint& operator+=(const modint& a){ if((x += a.x) >= mod) x -= mod; return *this; } constexpr modint operator+(const modint& a) const{ return modint(*this) += a; } constexpr modint& operator-=(const modint& a){ if((x -= a.x) < 0) x += mod; return *this; } constexpr modint operator-(const modint& a) const{ return modint(*this) -= a; } constexpr modint& operator*=(const modint& a){ (x *= a.x)%=mod; return *this; } constexpr modint operator*(const modint& a) const{ return modint(*this) *= a; } constexpr modint pow(unsigned long long pw) const{ modint res(1), comp(*this); while(pw){ if(pw&1) res *= comp; comp *= comp; pw >>= 1; } return res; } //以下、modが素数のときのみ constexpr modint inv() const{ if(x == 2) return (mod + 1) >> 1; return modint(*this).pow(mod - 2); } constexpr modint& operator/=(const modint &a){ (x *= a.inv().x)%=mod; return *this; } constexpr modint operator/(const modint &a) const{ return modint(*this) /= a; } constexpr bool sqrt(bool find_mini = false) { if(x == 0) return true; modint jge = this->pow((mod - 1)>>1); if(jge.x + 1 == mod) return false; if((mod&3) == 3){ *this = this->pow((mod + 1)>>2); }else{ int m = 0; modint c, t; if(mod == 998244353){ m = 23; c = 15311432; t = this->pow(119); *this = this->pow(60); }else{ ll q = mod - 1; modint z = 2; while(!(q&1)){q>>=1; ++m;} while(z.pow((mod-1)>>1).x == 1) z += 1; c = z.pow(q); t = this->pow(q); *this = this->pow((q+1)>>1); } while(t.x != 1){ modint cpy_t = t; int pw = m; while(cpy_t.x != 1){--pw; cpy_t *= cpy_t;} rep(i, pw-1) c *= c; (*this) *= c; c *= c; t *= c; m -= pw; } } if(find_mini) this->x = min(this->x, (ll)mod - this->x); return true; } }; #define mod1 998244353 using mint = modint; ostream& operator<<(ostream& os, const mint& a){ os << a.x; return os; } using vm = vector; class NTT{ static int root; static vector id; static void make_bit_reverse(int n){ //n must be 2^k int sz = id.size(), _chk = n; while(_chk > 1){assert(!(_chk&1)); _chk>>=1;} if(n > sz) { id.resize(n); while (sz < n) { rep(i, sz) { id[i] <<= 1; id[i | sz] = id[i] | 1; } sz <<= 1; } } if(n < sz){ int k = 1; while(n * k < sz) k <<= 1; for(int i = 1, cpy = k; cpy < sz; ++i, cpy += k) id[i] = id[cpy]; id.resize(n); } } static void dft(vm &f, bool inv, int n = INT_MAX){ if(n > (int)f.size()) n = f.size(); make_bit_reverse(n); rep(i, n) if (i > id[i]) swap(f[i], f[id[i]]); int l{1}; for (int len = 1; len < n; ++l, len <<= 1) { mint root_diff = mint(root).pow(inv ? (mod1 - 1) - ((mod1 - 1)>>l) : ((mod1 - 1)>>l)); int len2 = len << 1; for (int i = 0; i < n; i += len2) { mint z = 1; reps(j, i, i + len) { mint z_f = z * f[j + len]; f[j + len] = f[j] - z_f; f[j] += z_f; z *= root_diff; } } } if(inv) { mint n_inv = mint(n).inv(); rep(i, n) f[i] *= n_inv; } } public: static void dft_2D(int n, int m, vector &a, bool inv){ //簡単に、書き換える形で //aがn×mサイズであることや、n,mが2冪であることは仮定 rep(i, n) dft(a[i], inv); rep(j, m){ vm temp(n); rep(i, n) temp[i] = a[i][j]; dft(temp, inv); rep(i, n) a[i][j] = temp[i]; } } static vm convolution(vm g, vm h){ int sz = g.size() + h.size() - 1, n = 1; while(sz > n) n *= 2; g.resize(n); h.resize(n); dft(g, false); dft(h, false); rep(i, n) g[i] *= h[i]; dft(g, true); g.resize(sz); return g; } static vm simple_pow(vm &a, ll pw){ int sz = a.size(), n = 1; while(sz > n) n <<= 1; n <<= 1; vm res(n, 0), cpy(n, 0); res[0] = 1; copy(ALL(a), cpy.begin()); while(pw){ dft(cpy, false); if(pw&1){ dft(res, false); rep(i, n) res[i] *= cpy[i]; dft(res, true); reps(i, n / 2, n) res[i] = 0; } rep(i, n) cpy[i] *= cpy[i]; dft(cpy, true); reps(i, n / 2, n) cpy[i] = 0; pw >>= 1; } res.resize(sz); return res; } static vm inversion(vm f){ assert(f[0].x != 0); int n = 1, sz = 1, first_sz = f.size(); while(first_sz > n) n <<= 1; f.resize(n); vm g(n), cpy_f(n<<1), cpy_g(n<<1); g[0] = f[0].inv(); while(sz < n){ sz <<= 1; copy(f.begin(), f.begin() + sz, cpy_f.begin()); copy(g.begin(), g.begin() + sz, cpy_g.begin()); dft(cpy_f, false, sz<<1); dft(cpy_g, false, sz<<1); rep(i, sz<<1) cpy_f[i] *= cpy_g[i] * cpy_g[i]; dft(cpy_f, true, sz<<1); rep(i, sz) (g[i] += g[i])-= cpy_f[i]; } g.resize(first_sz); return g; } static void differential(vm &f){ int n = f.size(); rep(i, n - 1) f[i] = f[i + 1] * (i + 1); f[n-1] = 0; } static void integral(vm &f){ int n = f.size(); Rreps(i, n, 1) f[i] = f[i - 1]; f[0] = 0; mint fct(1); reps(i, 1, n){f[i] *= fct; fct*=i;} fct = fct.inv(); Rreps(i, n, 1){f[i] *= fct; fct*=i;} } static vm log(vm f){ assert(f[0].x == 1); vm res = inversion(f); differential(f); res = convolution(f, res); res.resize((int)f.size()); integral(res); return res; } static vm exp(vm f){ assert(f[0].x == 0); int n = 1, sz = 1, first_sz = f.size(); while(first_sz > n) n <<= 1; f.resize(n); vm g(1, 1), log_g; while(sz < n){ sz <<= 1; g.resize(sz); reps(i, sz>>1, sz) g[i] = 0; log_g = log(g); rep(i, sz) log_g[i] = f[i] - log_g[i]; log_g[0] += 1; g = convolution(g, log_g); } g.resize(first_sz); return g; } static vm pow(vm f, ll pw_int){ int n = f.size(), non_zero_id{0}; mint pw(pw_int); vm res(n, 0); if(pw_int == 0){ res[0] = 1; return res; } while(non_zero_id < n && f[non_zero_id].x == 0) ++non_zero_id; if((n + pw_int - 1) / pw_int <= non_zero_id) return res; mint d = f[non_zero_id], d_inv = d.inv(); rep(i, n - non_zero_id) f[i] = f[i + non_zero_id] * d.inv(); non_zero_id *= pw_int; d = d.pow(pw_int%(mod1 - 1)); f.resize(n - non_zero_id); res = log(f); for(auto &e : res) e *= pw; res = exp(res); res.resize(n); Rrep(i, n) res[i] = (i >= non_zero_id ? res[i - non_zero_id] * d : 0); return res; } static vm geometric_series(vm f){ assert(f[0].x == 0); f[0] = 1; reps(i, 1, (int)f.size()) if(f[i].x) f[i].set_raw(mod1 - f[i].x); return inversion(f); } static vm sqrt(vm f, bool &suc){ suc = true; int n = 1, sz = 1, first_sz = f.size(), non_zero_id{0}; vm g(1, 1); while(non_zero_id < first_sz && f[non_zero_id].x == 0){ ++non_zero_id; if(non_zero_id == first_sz) return vm(first_sz, 0); if(f[non_zero_id].x != 0){suc = false; return g;} ++non_zero_id; } if(non_zero_id >= first_sz)return vm(first_sz, 0); mint sq = f[non_zero_id], div = sq.inv(); if(!sq.sqrt(true)){suc = false; return g;} rep(i, first_sz - non_zero_id) f[i] = f[i+non_zero_id] * div; reps(i, first_sz - non_zero_id, first_sz) f[i] = 0; non_zero_id >>= 1; while(first_sz > n) n <<= 1; f.resize(n); vm cpy_f, g_inv; while(sz < n){ sz<<=1; g.resize(sz); cpy_f.resize(sz); g_inv = inversion(g); copy(f.begin(), f.begin() + sz, cpy_f.begin()); g_inv = convolution(cpy_f, g_inv); rep(i, sz) { g[i] += g_inv[i]; if(g[i].x&1) g[i].set_raw((g[i].x + mod1)>>1); else g[i].set_raw(g[i].x>>1); } } g.resize(first_sz); Rreps(i, first_sz, non_zero_id) g[i] = g[i - non_zero_id] * sq; rep(i, non_zero_id) g[i] = 0; return g; } /* 未完成 static vm composition(vm f, vm g){ assert(g[0].x == 0); int first_sz = min(f.size(), g.size()), n = 1, k = 0; while(first_sz > n) {n <<= 1; ++k;} { int lb = (1 << (k >> 1)) / k, ub = (1 << ((k + 1) >> 1)) + 1; while (ub - lb > 1) {int cen = (ub + lb) >> 1; (cen * cen * k <= n ? lb : ub) = cen;} rep(i, n){ chmax(lb, i + 1); if(g[i].x != 0) break; if(i == n - 1){ vm res(first_sz, 0); res[0] = f[0]; return res; } } k = 1; while(k < lb) k <<= 1; if(k > n) k = n; } vm p(n<<1), p_prime_inv(k), q(n<<1), qpow, res, temp(n<<1); vector dp(n); f.resize(n); g.resize(n); reps(i, first_sz, n) f[i] = g[i] = 0; copy(g.begin(), g.begin() + min(k, (int)g.size()), p.begin()); copy(g.begin(), g.begin() + min(k, (int)g.size()), p_prime_inv.begin()); copy(g.begin() + k, g.end(), q.begin()); dft(q, false); qpow = q; dft(p, false, k<<1); rep(i, n) dp[i].push_back(f[i]); auto calc = [&](vm &s, vm &t){ assert((int)t.size() <= k && (int)s.size() <= k); t.resize(k<<1); dft(t, false); rep(j, k<<1) t[j] *= p[j]; dft(t, true); rep(j, (int)s.size()) t[j] += s[j]; if(t.size() > n) t.resize(n); return t; }; for(int cmp = 1; cmp < n; cmp <<= 1){ for(int i = 0; i < n; i += (cmp<<1)){ dp[i] = calc(dp[i], dp[i + cmp]); dp[i + cmp].clear(); } if((cmp<<1) != n){ rep(i, k<<1) p[i] *= p[i]; dft(p, true, k<<1); if(k < n) k <<= 1; else {reps(i, n, n<<1) p[i] = 0;} dft(p, false, k<<1); } } res = dp[0]; res.resize(first_sz); k = p_prime_inv.size(); dp[0].resize(n<<1); dft(dp[0], false); differential(p_prime_inv); p_prime_inv = inversion(p_prime_inv); p_prime_inv.resize(n<<1); dft(p_prime_inv, false); for(int i = k; i < n; i += k){ rep(j, n<<1) dp[0][j] *= p_prime_inv[j]; dft(dp[0], true); mint i_inv = mint(i / k).inv(); rep(j, n) dp[0][j] *= i_inv; reps(j, n, n<<1) dp[0][j] = 0; differential(dp[0]); dft(dp[0], false); copy(dp[0].begin(), dp[0].end(), temp.begin()); rep(j, n<<1) { temp[j] *= qpow[j]; qpow[j] *= q[j]; } dft(temp, true); reps(j, i, first_sz) res[j] += temp[j-i]; dft(qpow, true); reps(j, n, n<<1) qpow[j] = 0; dft(qpow, false); } return res; } */ }; int NTT::root = 3; vector NTT::id(1, 0); const ll MAX_N = ll(4e+5) + 10; vm fact(MAX_N, mint(1)), fact_inv(MAX_N, mint(1)), n_inv(MAX_N, mint(1)); void makefact(){ mint tmp; reps(i,2,MAX_N) fact[i] = fact[i-1] * tmp.set_raw(i); fact_inv[MAX_N - 1] = fact[MAX_N - 1].inv(); Rreps(i, MAX_N - 1, 1){ fact_inv[i] = fact_inv[i + 1] * tmp.set_raw(i + 1); n_inv[i + 1] = fact[i] * fact_inv[i + 1]; } } mint nCm(ll n, ll m){ return fact[n] * fact_inv[n-m] * fact_inv[m]; } mint nCm_inv(ll n, ll m){ return fact[n-m] * fact[m] * fact_inv[n]; } int main(){ cin.tie(nullptr); ios::sync_with_stdio(false); makefact(); cin>>N>>M>>K; vm Kcolor_in_i(N - K + 1); mint res(0), mpow(1), M_C_K = nCm(M, K); rep(i, N - K + 1) Kcolor_in_i[i] = fact_inv[i + 1]; Kcolor_in_i = NTT::pow(Kcolor_in_i, K); rep(i, N - K + 1) Kcolor_in_i[i] *= fact[i + K]; Rreps(i, N + 1, K){ res += nCm(N, i) * mpow * Kcolor_in_i[i - K] * M_C_K; mpow *= M; } cout<