use std::io::Read; fn get_word() -> String { let stdin = std::io::stdin(); let mut stdin=stdin.lock(); let mut u8b: [u8; 1] = [0]; loop { let mut buf: Vec = Vec::with_capacity(16); loop { let res = stdin.read(&mut u8b); if res.unwrap_or(0) == 0 || u8b[0] <= b' ' { break; } else { buf.push(u8b[0]); } } if buf.len() >= 1 { let ret = String::from_utf8(buf).unwrap(); return ret; } } } #[allow(dead_code)] fn get() -> T { get_word().parse().ok().unwrap() } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl Default for ModInt { fn default() -> Self { Self::new_internal(0) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl ::std::fmt::Debug for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { let (mut a, mut b, _) = red(self.x, M::m()); if b < 0 { a = -a; b = -b; } write!(f, "{}/{}", a, b) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } // Finds the simplest fraction x/y congruent to r mod p. // The return value (x, y, z) satisfies x = y * r + z * p. fn red(r: i64, p: i64) -> (i64, i64, i64) { if r.abs() <= 10000 { return (r, 1, 0); } let mut nxt_r = p % r; let mut q = p / r; if 2 * nxt_r >= r { nxt_r -= r; q += 1; } if 2 * nxt_r <= -r { nxt_r += r; q -= 1; } let (x, z, y) = red(nxt_r, r); (x, y - q * z, z) } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 1_000_000_009; define_mod!(P, MOD); type MInt = mod_int::ModInt

; // Depends on MInt.rs fn fact_init(w: usize) -> (Vec, Vec) { let mut fac = vec![MInt::new(1); w]; let mut invfac = vec![0.into(); w]; for i in 1..w { fac[i] = fac[i - 1] * i as i64; } invfac[w - 1] = fac[w - 1].inv(); for i in (0..w - 1).rev() { invfac[i] = invfac[i + 1] * (i as i64 + 1); } (fac, invfac) } // FFT (in-place, verified as NTT only) // R: Ring + Copy // Verified by: https://judge.yosupo.jp/submission/53831 // Adopts the technique used in https://judge.yosupo.jp/submission/3153. mod fft { use std::ops::*; // n should be a power of 2. zeta is a primitive n-th root of unity. // one is unity // Note that the result is bit-reversed. pub fn fft(f: &mut [R], zeta: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let mut m = n; let mut base = zeta; unsafe { while m > 2 { m >>= 1; let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m); *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = w * (u - d); w = w * base; } r += 2 * m; } base = base * base; } if m > 1 { // m = 1 let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } } } } pub fn inv_fft(f: &mut [R], zeta_inv: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let zeta = zeta_inv; // inverse FFT let mut zetapow = Vec::with_capacity(20); { let mut m = 1; let mut cur = zeta; while m < n { zetapow.push(cur); cur = cur * cur; m *= 2; } } let mut m = 1; unsafe { if m < n { zetapow.pop(); let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } m = 2; } while m < n { let base = zetapow.pop().unwrap(); let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m) * w; *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = u - d; w = w * base; } r += 2 * m; } m *= 2; } } } } mod arbitrary_mod { use crate::mod_int::{self, ModInt}; use crate::fft; const MOD1: i64 = 1012924417; const MOD2: i64 = 1224736769; const MOD3: i64 = 1007681537; const G1: i64 = 5; const G2: i64 = 3; const G3: i64 = 3; define_mod!(P1, MOD1); define_mod!(P2, MOD2); define_mod!(P3, MOD3); fn zmod(mut a: i64, b: i64) -> i64 { a %= b; if a < 0 { a += b; } a } fn ext_gcd(mut a: i64, mut b: i64) -> (i64, i64, i64) { let mut x = 0; let mut y = 1; let mut u = 1; let mut v = 0; while a != 0 { let q = b / a; x -= q * u; std::mem::swap(&mut x, &mut u); y -= q * v; std::mem::swap(&mut y, &mut v); b -= q * a; std::mem::swap(&mut b, &mut a); } (b, x, y) } fn invmod(a: i64, b: i64) -> i64 { let x = ext_gcd(a, b).1; zmod(x, b) } // This function is ported from http://math314.hateblo.jp/entry/2015/05/07/014908 fn garner(mut mr: Vec<(i64, i64)>, mo: i64) -> i64 { mr.push((mo, 0)); let mut coffs = vec![1; mr.len()]; let mut constants = vec![0; mr.len()]; for i in 0..mr.len() - 1 { let v = zmod(mr[i].1 - constants[i], mr[i].0) * invmod(coffs[i], mr[i].0) % mr[i].0; assert!(v >= 0); for j in i + 1..mr.len() { constants[j] += coffs[j] * v % mr[j].0; constants[j] %= mr[j].0; coffs[j] = coffs[j] * mr[i].0 % mr[j].0; } } constants[mr.len() - 1] } // f *= g, g is destroyed fn convolution_friendly(a: &[i64], b: &[i64], gen: i64) -> Vec { let d = a.len(); let mut f = vec![ModInt::

::new(0); d]; let mut g = vec![ModInt::

::new(0); d]; for i in 0..d { f[i] = a[i].into(); g[i] = b[i].into(); } let zeta = ModInt::new(gen).pow((P::m() - 1) / d as i64); fft::fft(&mut f, zeta, ModInt::new(1)); fft::fft(&mut g, zeta, ModInt::new(1)); for i in 0..d { f[i] *= g[i]; } fft::inv_fft(&mut f, zeta.inv(), ModInt::new(1)); let inv = ModInt::new(d as i64).inv(); let mut ans = vec![0; d]; for i in 0..d { ans[i] = (f[i] * inv).x; } ans } // Precondition: 0 <= a[i], b[i] < mo pub fn arbmod_convolution(a: &[i64], b: &[i64], mo: i64, ret: &mut [i64]) { use crate::mod_int::Mod; let d = a.len(); assert!(d.is_power_of_two()); assert_eq!(d, b.len()); let x = convolution_friendly::(&a, &b, G1); let y = convolution_friendly::(&a, &b, G2); let z = convolution_friendly::(&a, &b, G3); let mut mr = [(0, 0); 3]; for i in 0..d { mr[0] = (P1::m(), x[i]); mr[1] = (P2::m(), y[i]); mr[2] = (P3::m(), z[i]); ret[i] = garner(mr.to_vec(), mo); } } pub fn arbmod_convolution_modint( a: &[ModInt

], b: &[ModInt

], ret: &mut [ModInt

]) { let mo = P::m(); unsafe { arbmod_convolution(std::mem::transmute(a), std::mem::transmute(b), mo, std::mem::transmute(ret)); } } } // Computes exp(f) mod x^{f.len()}. // Reference: https://arxiv.org/pdf/1301.5804.pdf // Complexity: O(n log n) fn fps_exp_arb( h: &[mod_int::ModInt

], fac: &[mod_int::ModInt

], invfac: &[mod_int::ModInt

], ) -> Vec> { use arbitrary_mod::*; let n = h.len(); assert!(n.is_power_of_two()); assert_eq!(h[0], 0.into()); let mut m = 1; let mut f = vec![mod_int::ModInt::new(0); n]; let mut g = vec![mod_int::ModInt::new(0); n]; let mut tmp = vec![mod_int::ModInt::new(0); n]; let mut tmp2 = vec![mod_int::ModInt::new(0); n]; f[0] = 1.into(); g[0] = 1.into(); // Adopts the technique used in https://judge.yosupo.jp/submission/3153 while m < n { // upheld invariants: f = exp(h) (mod x^m) // g = exp(-h) (mod x^(m/2)) // 2.a': g = 2g - fg^2 mod x^m if m > 1 { // The following can be dropped because the actual // computation was done in the previous iteration. // tmp_g[..m].copy_from_slice(&g[..m]); // fft::fft(&mut tmp_g[..m], zeta, 1.into()); arbmod_convolution_modint(&f[..m], &g[..m], &mut tmp[..m]); for v in &mut tmp[..m / 2] { *v = 0.into(); } for v in &mut tmp[m / 2..m] { *v = -*v; } arbmod_convolution_modint(&tmp[..m], &g[..m], &mut tmp2[..m]); g[m / 2..m].copy_from_slice(&tmp2[m / 2..m]); } // 2.b': q = h' mod x^(m-1) for i in 0..m - 1 { tmp[i] = h[i + 1] * (i + 1) as i64; } tmp[m - 1] = 0.into(); // 2.c': r = fq (mod x^m - 1) arbmod_convolution_modint(&tmp[..m], &f[..m], &mut tmp2[..m]); // 2.d' s = x(f' - r) mod (x^m - 1) for i in (0..m - 1).rev() { tmp2.swap(i, i + 1); } for i in 0..m { tmp[i] = f[i] * i as i64 - tmp2[i]; } // 2.e': t = gs mod x^m arbmod_convolution_modint(&tmp[..2 * m], &g[..2 * m], &mut tmp2[..2 * m]); // 2.f': u = (h mod x^2m - \int tx^(m-1)) / x^m for i in 0..m { tmp[i] = h[i + m] - tmp2[i] * fac[i + m - 1] * invfac[i + m]; } for v in &mut tmp[m..2 * m] { *v = 0.into(); } // 2.g': v = fu mod x^m arbmod_convolution_modint(&tmp[..2 * m], &f[..2 * m], &mut tmp2[..2 * m]); // 2.h': f += vx^m f[m..2 * m].copy_from_slice(&tmp2[..m]); // 2.i': m *= 2 m *= 2; } f } fn powmod(x: i64, mut e: i64, m: i64) -> i64 { let mut sum = 1; let mut cur = x % m; while e > 0 { if e % 2 != 0 { sum = sum * cur % m; } cur = cur * cur % m; e /= 2; } sum } /** * Calculates x s.t. x^2 = a (mod p) * p is prime * Verified by: CF #395 Div1-C * (http://codeforces.com/contest/763/submission/24380573) */ fn modsqrt(mut a: i64, p: i64) -> Option { a %= p; if a == 0 { return Some(0); } if p == 2 { return Some(a); } if powmod(a, (p - 1) / 2, p) != 1 { return None; } let mut b = 1; while powmod(b, (p - 1) / 2, p) == 1 { b += 1; } let mut e = 0; let mut m = p - 1; while m % 2 == 0 { m /= 2; e += 1; } let mut x = powmod(a, (m - 1) / 2, p); let mut y = a * (x * x % p) % p; x = x * a % p; let mut z = powmod(b, m, p); while y != 1 { let mut j = 0; let mut t = y; while t != 1 { j += 1; t = t * t % p; } assert!(j < e); z = powmod(z, 1 << (e - j - 1), p); x = x * z % p; z = z * z % p; y = y * z % p; e = j; } Some(x) } // https://yukicoder.me/problems/no/1080 (5) // N![x^K]exp(sqrt(-1) * (4x + 9x^2 + 16x^3 + …)) の実部 + 虚部が答え。mod (10^9 + 9) なので sqrt(-1) は存在するが、NTT friendly ではない。 // Tags: fps, ntt-unfriendly-mod, mod-sqrt fn main() { let im = modsqrt(MOD - 1, MOD).unwrap(); eprintln!("{}", im); let n: usize = get(); let mut p = 1; while p <= n { p *= 2; } let (fac, invfac) = fact_init(p + 1); let mut f = vec![MInt::new(0); p]; for i in 1..p { f[i] = MInt::new(im) * (i as i64 + 1) * (i as i64 + 1); } let pos = fps_exp_arb(&f, &fac, &invfac); for i in 1..p { f[i] = MInt::new(MOD - im) * (i as i64 + 1) * (i as i64 + 1); } let neg = fps_exp_arb(&f, &fac, &invfac); let inv2 = MInt::new(2).inv(); for i in 1..n + 1 { let re = pos[i] + neg[i]; let im = -(pos[i] - neg[i]) * MInt::new(im); println!("{}", (re + im) * fac[n] * inv2); } }