Problem Description

Let $\text{inv}_{\pi}$ denote the number of inversions of a permutation $\pi$. If the length of $\pi$ is $n$, then

$$\text{inv}_{\pi}=\sum\limits_{i=1}^{n}\sum\limits_{j=i+1}^{n}[\pi_i>\pi_j]$$

Given two positive integers $n,k$, and a permutation $\pi^\prime$, define the weight $\text{val}_{\pi}$ of a permutation $\pi$ of length $n$ as

$$\text{val}_{\pi}=\prod\limits_{i=1}^{n}\prod_{j=i+1}^{n}\pi_i^{[\pi_{i}>\pi_j]}$$

For $0\leq m\leq k$, compute

$$\sum\limits_{\pi}[\text{inv}_\pi=m]\text{val}_\pi$$

where $\pi$ is a permutation of length $n$.

Input Format

The first line contains two integers $n,k$.

Output Format

Output one line with $k+1$ integers, representing the answers modulo $998244353$.

3 3

1 5 15 18

Hint

Sample Explanation 1

$$\text{inv}_{(1,2,3)}=0,\text{inv}_{(1,3,2)}=1,\text{inv}_{(2,1,3)}=1,\text{inv}_{(2,3,1)}=2,\text{inv}_{(3,1,2)}=2,\text{inv}_{(3,2,1)}=3$$$$\text{val}_{(1,2,3)}=1,\text{val}_{(1,3,2)}=3,\text{val}_{(2,1,3)}=2,\text{val}_{(2,3,1)}=6,\text{val}_{(3,1,2)}=9,\text{val}_{(3,2,1)}=18$$

So when $m=0$ the answer is $1$, when $m=1$ it is $5$, when $m=2$ it is $15$, and when $m=3$ it is $18$.

Constraints

Subtask ID	Points	$n\leq$	$k\leq$
Subtask 1	$1$	$6$
Subtask 2	$12$	$40$
Subtask 3	$1$	$998244352$	$1$
Subtask 4	$16$		$10$
Subtask 5	$24$	$2\times 10^5$
Subtask 6	$46$	$998244352$

For $100\%$ of the testdata, $2\leq n\leq 998244352$, $1\leq k\leq \min(2\times 10^5,\frac{n(n-1)}{2})。$

Translated by ChatGPT 5