Problem:
Score : 700 points

Problem Statement

Takahashi throws N dice, each having K sides with all integers from 1 to K. The dice are NOT pairwise distinguishable.
For each i=2,3,...,2K, find the following value modulo 998244353:

The number of combinations of N sides shown by the dice such that the sum of no two different sides is i.

Note that the dice are NOT distinguishable, that is, two combinations are considered different when there exists an integer k such that the number of dice showing k is different in those two.

Constraints

1 \leq K \leq 2000

2 \leq N \leq 2000

K and N are integers.

Input

Input is given from Standard Input in the following format:

K N

Output

Print 2K-1 integers. The t-th of them (1\leq t\leq 2K-1) should be the answer for i=t+1.

Sample Input 1

3 3

Sample Output 1

7
7
4
7
7

For i=2, the combinations (1,2,2),(1,2,3),(1,3,3),(2,2,2),(2,2,3),(2,3,3),(3,3,3) satisfy the condition, so the answer is 7.

For i=3, the combinations (1,1,1),(1,1,3),(1,3,3),(2,2,2),(2,2,3),(2,3,3),(3,3,3) satisfy the condition, so the answer is 7.

For i=4, the combinations (1,1,1),(1,1,2),(2,3,3),(3,3,3) satisfy the condition, so the answer is 4.

Sample Input 2

4 5

Sample Output 2

36
36
20
20
20
36
36

Sample Input 3

6 1000

Sample Output 3

149393349
149393349
668669001
668669001
4000002
4000002
4000002
668669001
668669001
149393349
149393349

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 {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($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:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

const MOD: i64 = 998244353;
/// Refers to external ::MOD.
/// Verified by: https://beta.atcoder.jp/contests/arc099/submissions/2893648
mod mod_int {
    use ::MOD;
    use std::ops::*;
    #[derive(Copy, Clone, Debug, Hash, PartialEq, Eq, PartialOrd, Ord)]
    pub struct ModInt { pub x: i64 }
    impl ModInt {
        fn check_integrity(self) {
            debug_assert!(self.x >= 0);
            debug_assert!(self.x < MOD);
        }
        // x >= 0
        pub fn new<T: Into<i64>>(x: T) -> Self { ModInt { x: x.into() % MOD } }
        #[allow(dead_code)]
        pub fn mul_fast(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            ModInt { x: self.x * other.x % MOD }
        }
        #[allow(dead_code)]
        pub fn mul_slow(self, other: Self) -> Self {
            // Naive multiplication in order to avoid overflow
            self.check_integrity();
            other.check_integrity();
            let mut sum = ModInt::new(0);
            let mut cur = self;
            let mut e = other.x;
            if self.x < other.x {
                cur = other;
                e = self.x;
            }
            while e > 0 {
                if e % 2 == 1 {
                    sum = sum + cur;
                }
                cur = cur + cur;
                e /= 2;
            }
            sum
        }
        #[allow(dead_code)]
        pub fn pow(self, mut e: i64) -> Self {
            self.check_integrity();
            debug_assert!(e >= 0);
            let mut sum = ModInt::new(1);
            let mut cur = ModInt::new(self.x);
            while e > 0 {
                if e % 2 != 0 {
                    sum = sum * cur;
                }
                cur = cur * cur;
                e /= 2;
            }
            sum
        }
        #[allow(dead_code)]
        pub fn inv(self) -> Self { self.pow(MOD - 2) }
    }
    impl Add for ModInt {
        type Output = Self;
        fn add(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            let mut sum = self.x + other.x;
            if sum >= MOD { sum -= MOD; }
            ModInt { x: sum }
        }
    }
    impl Sub for ModInt {
        type Output = Self;
        fn sub(self, other: Self) -> Self {
            self.check_integrity();
            other.check_integrity();
            let mut sum = self.x - other.x;
            if sum < 0 { sum += MOD; }
            ModInt { x: sum }
        }
    }
    impl Mul for ModInt {
        type Output = Self;
        fn mul(self, other: Self) -> Self {
            self.mul_fast(other)
        }
    }
    impl AddAssign for ModInt {
        fn add_assign(&mut self, rhs: ModInt) { *self = *self + rhs; }
    }
    impl SubAssign for ModInt {
        fn sub_assign(&mut self, rhs: ModInt) { *self = *self - rhs; }
    }
    impl MulAssign for ModInt {
        fn mul_assign(&mut self, rhs: ModInt) { *self = *self * rhs; }
    }
    impl ::std::fmt::Display for ModInt {
        fn fmt(&self, f: &mut::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
} // mod mod_int
use mod_int::*;


fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($format:expr) => (write!(out,$format).unwrap());
        ($format:expr, $($args:expr),+) => (write!(out,$format,$($args),*).unwrap())
    }
    input!{
        k: usize,
        n: usize,
    }
    let mut dp = vec![ModInt::new(0); n + 1];
    dp[0] = ModInt::new(1);
    dp[1] = ModInt::new(1);
    for _ in 0 .. k - 1 {
        for j in 0 .. n {
            dp[j + 1] += dp[j];
        }
    }
    let mut ans = vec![ModInt::new(0); 2 * k + 1];
    for pl in 2 .. k + 2 {
        ans[pl] = dp[n];
        if pl % 2 == 1 {
            for j in (0 .. n).rev() {
                dp[j + 1] += dp[j];
            }
            for j in (0 .. n).rev() {
                dp[j + 1] -= dp[j];
            }
        }
    }
    for pl in k + 2 .. 2 * k + 1 {
        ans[pl] = ans[2 * k + 2 - pl];
    }
    for i in 2 .. 2 * k + 1 {
        puts!("{}\n", ans[i]);
    }
}

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();
}