Problem:
Score : 1900 points

Problem Statement

#nck {
width: 30px;
height: auto;
}

One day, Takahashi was given the following problem from Aoki:

You are given a tree with N vertices and an integer K. The vertices are numbered 1 through N. The edges are represented by pairs of integers (a_i, b_i).

For a set S of vertices in the tree, let f(S) be the minimum number of the vertices in a subtree of the given tree that contains all vertices in S.

There are ways to choose K vertices from the trees. For each of them, let S be the set of the chosen vertices, and find the sum of f(S) over all ways.

Since the answer may be extremely large, print it modulo 924844033(prime).

Since it was too easy for him, he decided to solve this problem for all K = 1,2,...,N.

Constraints

2 ≦ N ≦ 200,000

1 ≦ a_i, b_i ≦ N

The given graph is a tree.

Input

The input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
:
a_{N-1} b_{N-1}

Output

Print N lines. The i-th line should contain the answer to the problem where K=i, modulo 924844033.

Sample Input 1

3
1 2
2 3

Sample Output 1

3
7
3

The diagram above illustrates the case where K=2. The chosen vertices are colored pink, and the subtrees with the minimum number of vertices are enclosed by red lines.

Sample Input 2

4
1 2
1 3
1 4

Sample Output 2

4
15
13
4

Sample Input 3

7
1 2
2 3
2 4
4 5
4 6
6 7

Sample Output 3

7
67
150
179
122
45
7

Solution in Rust (1.15.1) (judge verdict: Accepted)

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::{Write, BufWriter};
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    }};

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

/// Verified by https://atcoder.jp/contests/arc093/submissions/3968098
mod mod_int {
    use std::ops::*;
    pub trait Mod: Copy { fn m() -> i64; }
    #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)]
    pub struct ModInt<M> { pub x: i64, phantom: ::std::marker::PhantomData<M> }
    impl<M: Mod> ModInt<M> {
        // 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<M: Mod, T: Into<ModInt<M>>> Add<T> for ModInt<M> {
        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<M: Mod, T: Into<ModInt<M>>> Sub<T> for ModInt<M> {
        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<M: Mod, T: Into<ModInt<M>>> Mul<T> for ModInt<M> {
        type Output = Self;
        fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) }
    }
    impl<M: Mod, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
        fn add_assign(&mut self, other: T) { *self = *self + other; }
    }
    impl<M: Mod, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
        fn sub_assign(&mut self, other: T) { *self = *self - other; }
    }
    impl<M: Mod, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
        fn mul_assign(&mut self, other: T) { *self = *self * other; }
    }
    impl<M: Mod> Neg for ModInt<M> {
        type Output = Self;
        fn neg(self) -> Self { ModInt::new(0) - self }
    }
    impl<M> ::std::fmt::Display for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
    impl<M> ::std::fmt::Debug for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
    impl<M: Mod> From<i64> for ModInt<M> {
        fn from(x: i64) -> Self { Self::new(x) }
    }
} // 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 = 924844033;
define_mod!(P, MOD);
type ModInt = mod_int::ModInt<P>;

// Depends on ModInt.rs
fn fact_init(w: usize) -> (Vec<ModInt>, Vec<ModInt>) {
    let mut fac = vec![ModInt::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)
}

fn dfs(v: usize, par: usize, g: &[Vec<usize>], freq: &mut [usize])
       -> usize /* size of subtree */ {
    let mut sz = 1;
    for &w in &g[v] {
        if w == par { continue; }
        sz += dfs(w, v, g, freq);
    }
    if par < g.len() { // v is not a root
        freq[sz] += 1;
        freq[g.len() - sz] += 1;
    }
    sz
}

/// FFT (in-place, verified as NTT only)
/// R: Ring + Copy
/// Verified by: https://codeforces.com/contest/1096/submission/47672373
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 should be multiplied by 1/sqrt(n).
    pub fn transform<R>(f: &mut [R], zeta: R, one: R)
        where R: Copy +
        Add<Output = R> +
        Sub<Output = R> +
        Mul<Output = R> {
        let n = f.len();
        assert!(n.is_power_of_two());
        {
            let mut i = 0;
            for j in 1 .. n - 1 {
                let mut k = n >> 1;
                loop {
                    i ^= k;
                    if k <= i { break; }
                    k >>= 1;
                }
                if j < i { f.swap(i, j); }
            }
        }
        let mut zetapow = Vec::new();
        {
            let mut m = 1;
            let mut cur = zeta;
            while m < n {
                zetapow.push(cur);
                cur = cur * cur;
                m *= 2;
            }
        }
        let mut m = 1;
        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[s];
                    let d = f[s + m] * w;
                    f[s] = u + d;
                    f[s + m] = u - d;
                    w = w * base;
                }
                r += 2 * m;
            }
            m *= 2;
        }
    }
}

use fft::*;

fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($($format:tt)*) => (write!(out,$($format)*).unwrap());
    }
    input! {
        n: usize,
        ab: [(usize1, usize1); n - 1],
    }
    let (fac, invfac) = fact_init(n + 1);
    let mut g = vec![vec![]; n];
    for &(a, b) in &ab {
        g[a].push(b);
        g[b].push(a);
    }
    let mut freq = vec![0; n + 1];
    dfs(0, n, &g, &mut freq);
    const N: usize = 1 << 19;
    let mut f = vec![ModInt::new(0); N];
    for i in 0..n + 1 {
        f[i] = fac[i] * freq[i] as i64;
    }
    let mut g = vec![ModInt::new(0); N]; // base = N / 2
    for i in 0..n + 1 {
        g[N / 2 - i] = invfac[i];
    }
    //eprintln!("f = {:?}", f);
    //eprintln!("g = {:?}", g);
    // zeta transform
    let one = ModInt::new(1);
    let gen = ModInt::new(5);
    let zeta = gen.pow((MOD - 1) / N as i64);
    let invzeta = zeta.inv();

    transform(&mut f, zeta, one);
    transform(&mut g, zeta, one);
    for i in 0..N { f[i] *= g[i]; }
    transform(&mut f, invzeta, one);
    transform(&mut g, invzeta, one);
    for i in 0..N {
        f[i] *= ModInt::new(N as i64).inv();
    }

    // whole
    let mut whole = vec![ModInt::new(0); n + 1];
    for k in 0..n + 1 {
        whole[k] = fac[n] * invfac[k] * invfac[n - k] * n as i64;
    }

    for k in 1..n + 1 {
        puts!("{}\n", whole[k] - f[N / 2 + k] * invfac[k]);
    }
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}