import std.conv, std.functional, std.range, std.stdio, std.string; import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons; import core.bitop; class EOFException : Throwable { this() { super("EOF"); } } string[] tokens; string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; } int readInt() { return readToken.to!int; } long readLong() { return readToken.to!long; } real readReal() { return readToken.to!real; } bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } } bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } } int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; } int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); } int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); } struct ModInt(int M_) { import std.conv : to; alias M = M_; int x; this(ModInt a) { x = a.x; } this(long a) { x = cast(int)(a % M); if (x < 0) x += M; } ref ModInt opAssign(long a) { return (this = ModInt(a)); } ref ModInt opOpAssign(string op)(ModInt a) { static if (op == "+") { x += a.x; if (x >= M) x -= M; } else static if (op == "-") { x -= a.x; if (x < 0) x += M; } else static if (op == "*") { x = cast(int)((cast(long)(x) * a.x) % M); } else static if (op == "/") { this *= a.inv(); } else static assert(false); return this; } ref ModInt opOpAssign(string op)(long a) { static if (op == "^^") { if (a < 0) return (this = inv()^^(-a)); ModInt t2 = this, te = ModInt(1); for (long e = a; e > 0; e >>= 1) { if (e & 1) te *= t2; t2 *= t2; } x = cast(int)(te.x); return this; } else return mixin("this " ~ op ~ "= ModInt(a)"); } ModInt inv() const { int a = x, b = M, y = 1, z = 0, t; for (; ; ) { t = a / b; a -= t * b; if (a == 0) { assert(b == 1 || b == -1); return ModInt(b * z); } y -= t * z; t = b / a; b -= t * a; if (b == 0) { assert(a == 1 || a == -1); return ModInt(a * y); } z -= t * y; } } ModInt opUnary(string op: "-")() const { return ModInt(-x); } ModInt opBinary(string op, T)(T a) const { return mixin("ModInt(this) " ~ op ~ "= a"); } ModInt opBinaryRight(string op)(long a) const { return mixin("ModInt(a) " ~ op ~ "= this"); } bool opCast(T: bool)() const { return (x != 0); } string toString() const { return x.to!string; } } enum MO = 10^^9 + 9; alias Mint = ModInt!MO; enum Mint I = 569522298; // a^-1 (mod m) long modInv(long a, long m) in { assert(m > 0, "modInv: m > 0 must hold"); } do { long b = m, x = 1, y = 0, t; for (; ; ) { t = a / b; a -= t * b; if (a == 0) { assert(b == 1 || b == -1, "modInv: gcd(a, m) != 1"); if (b == -1) y = -y; return (y < 0) ? (y + m) : y; } x -= t * y; t = b / a; b -= t * a; if (b == 0) { assert(a == 1 || a == -1, "modInv: gcd(a, m) != 1"); if (a == -1) x = -x; return (x < 0) ? (x + m) : x; } y -= t * x; } } // M: prime, G: primitive root class Fft(int M_, int G, int K) { import std.algorithm : reverse; import std.traits : isIntegral; alias M = M_; // 1, 1/4, 1/8, 3/8, 1/16, 5/16, 3/16, 7/16, ... int[] gs; this() { static assert(2 <= K && K <= 30, "Fft: 2 <= K <= 30 must hold"); static assert(!((M - 1) & ((1 << K) - 1)), "Fft: 2^K | M - 1 must hold"); gs = new int[1 << (K - 1)]; gs[0] = 1; long g2 = G, gg = 1; for (int e = (M - 1) >> K; e; e >>= 1) { if (e & 1) gg = (gg * g2) % M; g2 = (g2 * g2) % M; } gs[1 << (K - 2)] = cast(int)(gg); for (int l = 1 << (K - 2); l >= 2; l >>= 1) { gs[l >> 1] = cast(int)((cast(long)(gs[l]) * gs[l]) % M); } assert((cast(long)(gs[1]) * gs[1]) % M == M - 1, "Fft: g^(2^(K-1)) == -1 (mod M) must hold"); for (int l = 2; l <= 1 << (K - 2); l <<= 1) { foreach (i; 1 .. l) { gs[l + i] = cast(int)((cast(long)(gs[l]) * gs[i]) % M); } } } void fft(int[] xs) const { const n = cast(int)(xs.length); assert(!(n & (n - 1)), "Fft.fft: |xs| must be a power of two"); assert(n <= 1 << K, "Fft.fft: |xs| <= 2^K must hold"); for (int l = n; l >>= 1; ) { foreach (i; 0 .. (n >> 1) / l) { const(long) g = gs[i]; foreach (j; (i << 1) * l .. (i << 1 | 1) * l) { const t = cast(int)((g * xs[j + l]) % M); if ((xs[j + l] = xs[j] - t) < 0) xs[j + l] += M; if ((xs[j] += t) >= M) xs[j] -= M; } } } } void invFft(int[] xs) const { const n = cast(int)(xs.length); assert(!(n & (n - 1)), "Fft.invFft: |xs| must be a power of two"); assert(n <= 1 << K, "Fft.invFft: |xs| <= 2^K must hold"); for (int l = 1; l < n; l <<= 1) reverse(xs[l .. l << 1]); for (int l = 1; l < n; l <<= 1) { foreach (i; 0 .. (n >> 1) / l) { const(long) g = gs[i]; foreach (j; (i << 1) * l .. (i << 1 | 1) * l) { int t = cast(int)((g * (xs[j] - xs[j + l])) % M); if (t < 0) t += M; if ((xs[j] += xs[j + l]) >= M) xs[j] -= M; xs[j + l] = t; } } } } T[] convolute(T)(inout(T)[] as, inout(T)[] bs) const if (isIntegral!T) { const na = cast(int)(as.length), nb = cast(int)(bs.length); int n, invN = 1; for (n = 1; n < na + nb - 1; n <<= 1) { invN = ((invN & 1) ? (invN + M) : invN) >> 1; } auto xs = new int[n], ys = new int[n]; foreach (i; 0 .. na) if ((xs[i] = cast(int)(as[i] % M)) < 0) xs[i] += M; foreach (i; 0 .. nb) if ((ys[i] = cast(int)(bs[i] % M)) < 0) ys[i] += M; fft(xs); fft(ys); foreach (i; 0 .. n) { xs[i] = cast(int)((((cast(long)(xs[i]) * ys[i]) % M) * invN) % M); } invFft(xs); auto cs = new T[na + nb - 1]; foreach (i; 0 .. na + nb - 1) cs[i] = cast(T)(xs[i]); return cs; } ModInt!M[] convolute(inout(ModInt!M)[] as, inout(ModInt!M)[] bs) const { const na = cast(int)(as.length), nb = cast(int)(bs.length); int n, invN = 1; for (n = 1; n < na + nb - 1; n <<= 1) { invN = ((invN & 1) ? (invN + M) : invN) >> 1; } auto xs = new int[n], ys = new int[n]; foreach (i; 0 .. na) xs[i] = as[i].x; foreach (i; 0 .. nb) ys[i] = bs[i].x; fft(xs); fft(ys); foreach (i; 0 .. n) { xs[i] = cast(int)((((cast(long)(xs[i]) * ys[i]) % M) * invN) % M); } invFft(xs); auto cs = new ModInt!M[na + nb - 1]; foreach (i; 0 .. na + nb - 1) cs[i].x = xs[i]; return cs; } int[] convolute(int M1)(inout(ModInt!M1)[] as, inout(ModInt!M1)[] bs) const if (M != M1) { const na = cast(int)(as.length), nb = cast(int)(bs.length); int n, invN = 1; for (n = 1; n < na + nb - 1; n <<= 1) { invN = ((invN & 1) ? (invN + M) : invN) >> 1; } auto xs = new int[n], ys = new int[n]; foreach (i; 0 .. na) xs[i] = as[i].x; foreach (i; 0 .. nb) ys[i] = bs[i].x; fft(xs); fft(ys); foreach (i; 0 .. n) { xs[i] = cast(int)((((cast(long)(xs[i]) * ys[i]) % M) * invN) % M); } invFft(xs); return xs[0 .. na + nb - 1]; } int[] square(int M1)(inout(ModInt!M1)[] as) const if (M != M1) { const na = cast(int)(as.length); int n, invN = 1; for (n = 1; n < na + na - 1; n <<= 1) { invN = ((invN & 1) ? (invN + M) : invN) >> 1; } auto xs = new int[n]; foreach (i; 0 .. na) xs[i] = as[i].x; fft(xs); foreach (i; 0 .. n) { xs[i] = cast(int)((((cast(long)(xs[i]) * xs[i]) % M) * invN) % M); } invFft(xs); return xs[0 .. na + na - 1]; } } enum FFT_K = 20; alias Fft3_0 = Fft!(1045430273, 3, FFT_K); // 2^20 997 + 1 alias Fft3_1 = Fft!(1051721729, 6, FFT_K); // 2^20 1003 + 1 alias Fft3_2 = Fft!(1053818881, 7, FFT_K); // 2^20 1005 + 1 enum long FFT_INV01 = modInv(Fft3_0.M, Fft3_1.M); enum long FFT_INV012 = modInv(cast(long)(Fft3_0.M) * Fft3_1.M, Fft3_2.M); Fft3_0 FFT3_0; Fft3_1 FFT3_1; Fft3_2 FFT3_2; void initFft3() { FFT3_0 = new Fft3_0; FFT3_1 = new Fft3_1; FFT3_2 = new Fft3_2; } ModInt!M[] convolute(int M)(inout(ModInt!M)[] as, inout(ModInt!M)[] bs) { const cs0 = FFT3_0.convolute(as, bs); const cs1 = FFT3_1.convolute(as, bs); const cs2 = FFT3_2.convolute(as, bs); auto cs = new ModInt!M[cs0.length]; foreach (i; 0 .. cs0.length) { long d0 = cs0[i] % Fft3_0.M; long d1 = (FFT_INV01 * (cs1[i] - d0)) % Fft3_1.M; if (d1 < 0) d1 += Fft3_1.M; long d2 = (FFT_INV012 * ((cs2[i] - d0 - Fft3_0.M * d1) % Fft3_2.M)) % Fft3_2.M; if (d2 < 0) d2 += Fft3_2.M; cs[i] = (d0 + Fft3_0.M * d1 + ((cast(long)(Fft3_0.M) * Fft3_1.M) % M) * d2) % M; } return cs; } ModInt!M[] Square(int M)(inout(ModInt!M)[] as) { const cs0 = FFT3_0.square(as); const cs1 = FFT3_1.square(as); const cs2 = FFT3_2.square(as); auto cs = new ModInt!M[cs0.length]; foreach (i; 0 .. cs0.length) { long d0 = cs0[i] % Fft3_0.M; long d1 = (FFT_INV01 * (cs1[i] - d0)) % Fft3_1.M; if (d1 < 0) d1 += Fft3_1.M; long d2 = (FFT_INV012 * ((cs2[i] - d0 - Fft3_0.M * d1) % Fft3_2.M)) % Fft3_2.M; if (d2 < 0) d2 += Fft3_2.M; cs[i] = (d0 + Fft3_0.M * d1 + ((cast(long)(Fft3_0.M) * Fft3_1.M) % M) * d2) % M; } return cs; } struct Poly { Mint[] x; this(Poly f) { x = f.x.dup; } this(const(Poly) f) { x = f.x.dup; } this(int n) { x = new Mint[n]; } this(const(Mint)[] x) { this.x = x.dup; } this(const(long)[] x) { this.x.length = x.length; foreach (i; 0 .. x.length) this.x[i] = Mint(x[i]); } int size() const { return cast(int)(x.length); } Poly take(int n) const { return Poly(x[0 .. min(max(n, 1), $)]); } ref Poly opAssign(const(Mint)[] x) { this.x = x.dup; return this; } ref Poly opAssign(const(long)[] x) { this.x.length = x.length; foreach (i; 0 .. x.length) this.x[i] = Mint(x[i]); return this; } Mint opIndex(int i) const { return x[i]; } ref Mint opIndex(int i) { return x[i]; } ref Poly opOpAssign(string op)(const(Poly) f) { static if (op == "+") { if (size() < f.size()) x.length = f.size(); foreach (i; 0 .. f.size()) this[i] += f[i]; return this; } else static if (op == "-") { if (size() < f.size()) x.length = f.size(); foreach (i; 0 .. f.size()) this[i] -= f[i]; return this; } else static if (op == "*") { // TODO: FFT /* Poly g = Poly(size() + f.size() - 1); foreach (i; 0 .. size()) foreach (j; 0 .. f.size()) { g[i + j] += this[i] * f[j]; } this = g; return this; */ return this = Poly(convolute!MO(x, f.x)); } else { static assert(false); } } ref Poly opOpAssign(string op)(Mint a) if (op == "*") { foreach (i; 0 .. size()) this[i] *= a; return this; } Poly opBinary(string op, T)(T a) const { return mixin("Poly(this) " ~ op ~ "= a"); // Poly f = Poly(this); // mixin("f " ~ op ~ "= a;"); // return f; } Poly opBinaryRight(string op)(Mint a) const if (op == "*") { return this * a; } Poly opUnary(string op)() const if (op == "-") { return this * Mint(-1); } Poly square(int n) const { // TODO: FFT /* Poly f = Poly(n); foreach (i; 0 .. min(size(), (n + 1) / 2)) { f[i + i] += this[i] * this[i]; foreach (j; i + 1 .. min(size(), n - i)) { f[i + j] += Mint(2) * this[i] * this[j]; } } return f; */ return Poly(Square!MO(x)); } Poly inv(int n) const { // TODO: fft assert(this[0].x != 0); Poly f = Poly(n); f[0] = this[0].inv(); foreach (i; 1 .. n) { foreach (j; 1 .. min(size(), i + 1)) { f[i] -= this[j] * f[i - j]; } f[i] *= f[0]; } return f; } Poly differential() const { Poly f = Poly(max(size() - 1, 1)); foreach (i; 1 .. size()) f[i - 1] = Mint(i) * this[i]; return f; } Poly integral() const { Poly f = Poly(size() + 1); foreach (i; 0 .. size()) f[i + 1] = Mint(i + 1).inv() * this[i]; return f; } Poly exp(int n) const { assert(this[0].x == 0); const d = differential(); Poly f = [1], g = [1]; for (int m = 1; m < n; m <<= 1) { g = g + g - (f * g.square(m)).take(m); Poly h = d.take(m - 1); h += (g * (f.differential() - f * h)).take(2 * m - 1); f += (f * (take(2 * m) - h.integral())).take(2 * m); } return f.take(n); } } enum Poly1 = Poly([1]); enum PolyQ = Poly([0, 1]); void main() { initFft3; debug { // N = 3 auto sums = new Mint[][](3 + 1, 9 + 1); sums[0][0] += 6 * 1^^2; foreach (a; 1 .. 3 + 1) { sums[1][a] += 6 * (a + 1)^^2; } // foreach (a; 1 .. 3 + 1) foreach (b; a .. 3 + 1) { foreach (a; 1 .. 3 + 1) foreach (b; 1 .. 3 + 1) { sums[2][a + b] += 3 * ((a + 1) * (b + 1))^^2; } // foreach (a; 1 .. 3 + 1) foreach (b; a .. 3 + 1) foreach (c; b .. 3 + 1) { foreach (a; 1 .. 3 + 1) foreach (b; 1 .. 3 + 1) foreach (c; 1 .. 3 + 1) { sums[3][a + b + c] += 1 * ((a + 1) * (b + 1) * (c + 1))^^2; } foreach (m; 0 .. 3 + 1) { writeln(sums[m]); } foreach (k; 0 .. 9 + 1) { writeln(sums[0][k] + sums[1][k] - sums[2][k] - sums[3][k]); } } try { for (; ; ) { const N = readInt(); Mint fac = 1; foreach (i; 1 .. N + 1) { fac *= i; } auto f = Poly(N + 1); foreach (i; 1 .. N + 1) { f[i] = 1L * (i + 1) * (i + 1); } auto g0 = (f * I).exp(N + 1); auto g1 = (f * -I).exp(N + 1); auto re = (g0 + g1) * Mint(2).inv; auto im = (g0 - g1) * Mint(2 * I).inv; auto ans = (re + im) * fac; foreach (i; 1 .. N + 1) { writeln(ans[i]); } } } catch (EOFException e) { } }