// >>> TEMPLATES #include using namespace std; using ll = long long; using ld = long double; using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; #define int ll #define double ld #define rep(i,n) for (int i = 0; i < (int)(n); i++) #define rep1(i,n) for (int i = 1; i <= (int)(n); i++) #define repR(i,n) for (int i = (int)(n)-1; i >= 0; i--) #define rep1R(i,n) for (int i = (int)(n); i >= 1; i--) #define loop(i,a,B) for (int i = a; i B; i++) #define loopR(i,a,B) for (int i = a; i B; i--) #define all(x) (x).begin(), (x).end() #define allR(x) (x).rbegin(), (x).rend() #define pb push_back #define eb emplace_back #define mp make_pair #define fst first #define snd second template auto constexpr inf = numeric_limits::max()/2-1; auto constexpr INF32 = inf; auto constexpr INF64 = inf; auto constexpr INF = inf; #ifdef LOCAL #include "debug.hpp" #define dump(...) cerr << "[" << setw(3) << __LINE__ << ":" << __FUNCTION__ << "] ", dump_impl(#__VA_ARGS__, __VA_ARGS__) #define say(x) cerr << "[" << __LINE__ << ":" << __FUNCTION__ << "] " << x << endl #define debug if (1) #else #define dump(...) (void)(0) #define say(x) (void)(0) #define debug if (0) #endif template using pque_max = priority_queue; template using pque_min = priority_queue, greater >; template ::value>::type> ostream& operator<<(ostream& os, T const& v) { bool f = true; for (auto const& x : v) os << (f ? "" : " ") << x, f = false; return os; } template ::value>::type> istream& operator>>(istream& is, T &v) { for (auto& x : v) is >> x; return is; } template ostream& operator<<(ostream& os, pair const& p) { return os << "(" << p.first << ", " << p.second << ")"; } template istream& operator>>(istream& is, pair& p) { return is >> p.first >> p.second; } struct IOSetup { IOSetup() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); } } iosetup; template struct FixPoint : private F { constexpr FixPoint(F&& f) : F(forward(f)) {} template constexpr auto operator()(T&&... x) const { return F::operator()(*this, forward(x)...); } }; struct MakeFixPoint { template constexpr auto operator|(F&& f) const { return FixPoint(forward(f)); } }; #define MFP MakeFixPoint()| #define def(name, ...) auto name = MFP [&](auto &&name, __VA_ARGS__) template struct vec_impl { using type = vector::type>; template static type make_v(size_t n, U&&... x) { return type(n, vec_impl::make_v(forward(x)...)); } }; template struct vec_impl { using type = T; static type make_v(T const& x = {}) { return x; } }; template using vec = typename vec_impl::type; template auto make_v(Args&&... args) { return vec_impl::make_v(forward(args)...); } template void quit(T const& x) { cout << x << endl; exit(0); } template constexpr bool chmin(T& x, U const& y) { if (x > y) { x = y; return true; } return false; } template constexpr bool chmax(T& x, U const& y) { if (x < y) { x = y; return true; } return false; } template constexpr auto sumof(It b, It e) { return accumulate(b,e,typename iterator_traits::value_type{}); } template int sz(T const& x) { return x.size(); } template int lbd(C const& v, T const& x) { return lower_bound(v.begin(), v.end(), x)-v.begin(); } template int ubd(C const& v, T const& x) { return upper_bound(v.begin(), v.end(), x)-v.begin(); } template int ppt(C const& v, F f) { return partition_point(v.begin(), v.end(), f)-v.begin(); } // <<< // >>> FPS template struct FormalPowerSeries : NTT, vector { using mint = typename NTT::modint; using NTT::conv; using vector::vector; // inherit constructors using FPS = FormalPowerSeries; FormalPowerSeries() : vector() {} FormalPowerSeries(vector const& v) : vector(v) {} FormalPowerSeries(mint const& x) : vector({x}) {} mint get(int i) const { assert(i >= 0); if (i < (int)this->size()) return (*this)[i]; else return 0; } bool operator==(FPS const& r) const { const int n = min(this->size(), r.size()); rep (i,n) { if ((*this)[i] != r[i]) return false; } for (int i = n; i < (int)this->size(); ++i) { if ((*this)[i] != mint(0)) return false; } for (int i = n; i < (int)r.size(); ++i) { if (r[i] != mint(0)) return false; } return true; } bool operator!=(FPS const& r) const { return !((*this) == r); } FPS operator+(FPS const& r) const { return FPS(*this) += r; } FPS operator-(FPS const& r) const { return FPS(*this) -= r; } FPS& operator+=(FPS const& r) { if (r.size() > this->size()) this->resize(r.size()); rep (i,r.size()) (*this)[i] += r[i]; return *this; } FPS& operator-=(FPS const& r) { if (r.size() > this->size()) this->resize(r.size()); rep (i,r.size()) (*this)[i] -= r[i]; return *this; } FPS operator*(FPS const& r) const { if (this->empty() || r.empty()) return {}; return conv(*this,r); } FPS& operator*=(FPS const& r) { return *this = *this * r; } friend FPS operator+(mint const& x, FPS const& f) { return FPS{x}+f; } friend FPS operator-(mint const& x, FPS const& f) { return FPS{x}-f; } friend FPS operator*(mint const& x, FPS const& f) { return FPS{x}*f; } friend FPS operator+(FPS const& f, mint const& x) { return f+FPS{x}; } friend FPS operator-(FPS const& f, mint const& x) { return f-FPS{x}; } friend FPS operator*(FPS const& f, mint const& x) { return f*FPS{x}; } FPS take(int sz) const { FPS ret(this->begin(), this->begin() + min(this->size(),sz)); ret.resize(sz); return ret; } FPS inv(int sz = -1) const { assert(this->size()); assert((*this)[0] != mint(0)); if (sz < 0) sz = this->size(); FPS ret = { mint(1)/(*this)[0] }; for (int i = 1; i < sz; i <<= 1) { ret = ret + ret - ret*ret*take(i<<1); ret.resize(i<<1); } ret.resize(sz); return ret; } FPS diff() const { FPS ret(max(0,this->size()-1)); rep (i,ret.size()) ret[i] = (*this)[i+1]*mint(i+1); return ret; } FPS integral() const { FPS ret(this->size()+1); ret[0] = 0; rep (i,this->size()) ret[i+1] = (*this)[i]/mint(i+1); return ret; } FPS log(int sz = -1) const { assert(this->size()); assert((*this)[0] == mint(1)); if (sz < 0) sz = this->size(); return (diff()*inv(sz)).take(sz-1).integral(); } // FPS log(int sz = -1) const { // assert(this->size()); assert((*this)[0] == mint(1)); // if (sz < 0) sz = this->size(); // auto ret = diff()*inv(sz); // ret.resize(sz); // for (int i = sz-1; i > 0; --i) ret[i] = ret[i-1]/mint(i); // ret[0] = 0; // return ret; // } FPS exp(int sz = -1) const { FPS ret = {mint(1)}; if (this->empty()) return ret; assert((*this)[0] == mint(0)); if (sz < 0) sz = this->size(); for (int i = 1; i < sz; i <<= 1) { ret *= take(i<<1) + mint(1) - ret.log(i<<1); ret.resize(i<<1); } ret.resize(sz); return ret; } FPS pow(int64_t k, int sz = -1) const { if (sz < 0) sz = this->size(); int deg = 0; while (deg < sz && (*this)[deg] == mint(0)) ++deg; assert(k >= 0 || deg == 0); auto c = mint(1)/(*this)[deg]; FPS ret(sz-deg); rep (i,sz-deg) ret[i] = (*this)[deg+i]*c; ret = (ret.log()*k).exp() * (*this)[deg].pow(k); ret.resize(sz); for (int i = sz-1; i >= 0; --i) { int j = i-deg*k; ret[i] = (j >= 0 ? ret[j] : mint(0)); } return ret; } mint eval(mint x) const; }; // <<< // >>> NTT template struct NTT { using modint = ModInt; static constexpr int64_t mod = ModInt::mod, gen = g, max_lg = __builtin_ctzll(mod-1); // mod:prime, g:primitive root static_assert(mod > 0 && g > 0 && max_lg > 0, ""); using arr_t = array; static arr_t ws,iws; static void init() { static bool built = false; if (built) return; for (int i = 0; i <= max_lg; i++) { ws[i] = -ModInt(g).pow((mod-1)>>(i+2)); iws[i] = ModInt(1)/ws[i]; } built = true; } static void ntt(ModInt a[], int lg) { for (int b = lg-1; b >= 0; b--) { ModInt w = 1; for (int i = 0, k = 0; i < (1< static vector conv(vector const& a, vector const& b) { if (a.empty() || b.empty()) return {}; init(); const int s = a.size() + b.size() - 1, lg = __lg(2*s-1); assert(lg <= max_lg); vector aa(1< bb(1< static vector conv(vector const& a) { if (a.empty()) return {}; init(); const int s = a.size()*2 - 1, lg = __lg(2*s-1); assert(lg <= max_lg); vector aa(1< typename NTT::arr_t NTT::ws; template typename NTT::arr_t NTT::iws; // <<< // >>> modint template struct modint { using u32 = uint32_t; using u64 = uint64_t; using i64 = int64_t; using M = modint; static constexpr u32 mul_inv(u32 n, int e = 5, u32 x = 1) { return e == 0 ? x : mul_inv(n, e-1, x*(2-x*n)); } static constexpr u32 mod = MOD; static constexpr u32 nn = mul_inv(MOD); static constexpr u32 r2 = -u64(MOD) % MOD; u32 x; constexpr modint(i64 x = 0) : x(reduce(((x%=MOD)<0 ? x+MOD : x)*r2)) {} static constexpr u32 reduce(u64 w) { return u32(w >> 32) + MOD - i64((u64(u32(w) * nn) * MOD) >> 32); } constexpr i64 val() const { i64 r = reduce(x); if (r >= MOD) r -= MOD; return r; } constexpr explicit operator i64() const { return val(); } constexpr bool operator==(M p) const { return val() == p.val(); } constexpr bool operator!=(M p) const { return val() != p.val(); } constexpr M operator+() const { return *this; } constexpr M operator-() const { M r; r.x = x ? i64(2*MOD)-x : 0; return r; } constexpr M &operator+=(M p) { i64 t = x; if (((t += p.x) -= 2*MOD) < 0) t += 2*MOD; x = t; return *this; } constexpr M &operator-=(M p) { return *this += -p; } constexpr M &operator*=(M p) { x = reduce(u64(x)*p.x); return *this; } constexpr M &operator/=(M p) { *this *= p.inv(); return *this; } constexpr M operator+(M p) const { return M(*this) += p; } constexpr M operator-(M p) const { return M(*this) -= p; } constexpr M operator*(M p) const { return M(*this) *= p; } constexpr M operator/(M p) const { return M(*this) /= p; } friend constexpr M operator+(i64 x, M y) { return M(x)+y; } friend constexpr M operator-(i64 x, M y) { return M(x)-y; } friend constexpr M operator*(i64 x, M y) { return M(x)*y; } friend constexpr M operator/(i64 x, M y) { return M(x)/y; } constexpr M inv() const { return pow(MOD - 2); } constexpr M pow(i64 n) const { if (n < 0) return inv().pow(-n); M v = *this, r = 1; for (; n > 0; n >>= 1, v *= v) if (n&1) r *= v; return r; } friend ostream &operator<<(ostream &os, M p) { return os << p.val(); } friend istream &operator>>(istream &is, M &a) { u32 t; is >> t; a = t; return is; } #ifdef LOCAL friend string to_s(M p) { return to_s(p.val(), MOD); } #endif }; // <<< // >>> garner, convolution for arbitray mod template struct NTT_arb_mod { using modint = ModInt; using ntt1 = NTT<::modint< 167772161>, 3>; // mod-1 = 5<<25 using ntt2 = NTT<::modint< 469762049>, 3>; // mod-1 = 7<<26 using ntt3 = NTT<::modint<1224736769>, 3>; // mod-1 = 73<<24 //using ntt3 = NTT, 3>; // mod-1 = 119<<23 // https://math314.hateblo.jp/entry/2015/05/07/014908 static constexpr ModInt garner(ntt1::modint x, ntt2::modint y, ntt3::modint z) { using ll = int64_t; const ll xx = (ll)x, yy = (ll)y, zz = (ll)z; constexpr ll m1 = ntt1::mod, m2 = ntt2::mod; constexpr auto m1_inv_m2 = ntt2::modint(m1).inv(); constexpr auto m12_inv_m3 = ntt3::modint(m1 * m2).inv(); constexpr auto m12_mod = ModInt(m1 * m2); auto v1 = (yy - xx) * m1_inv_m2; auto v2 = (zz - (xx + m1 * v1.val())) * m12_inv_m3; return xx + m1 * v1.val() + m12_mod * ModInt(v2.val()); } static vector conv(vector const& a, vector const& b) { auto x = ntt1::conv(a, b); auto y = ntt2::conv(a, b); auto z = ntt3::conv(a, b); vector ret(x.size()); rep (i,x.size()) ret[i] = garner(x[i],y[i],z[i]); return ret; } static vector conv(vector const& a) { auto x = ntt1::conv(a); auto y = ntt2::conv(a); auto z = ntt3::conv(a); vector ret(x.size()); rep (i,x.size()) ret[i] = garner(x[i],y[i],z[i]); return ret; } }; // <<< constexpr int64_t MOD = 1e9+9; using mint = modint; //using FPS = FormalPowerSeries>; using FPS = FormalPowerSeries>; const mint I = mint(13).pow((MOD-1)/4); int32_t main() { int n; cin >> n; mint fact = 1; rep1 (i,n) fact *= i; FPS f_plus(n+1), f_minus(n+1); rep1 (i,n) f_plus[i] = (i+1)*(i+1)*I; rep1 (i,n) f_minus[i] = (i+1)*(i+1)*(-I); f_plus = f_plus.exp(); f_minus = f_minus.exp(); auto real = (f_plus + f_minus) * mint(2).inv(); auto imag = (f_plus - f_minus) * mint(2*I).inv(); auto ans = (real + imag) * fact; dump(ans); rep1 (k,n) cout << ans[k] << "\n"; }