Description

You are given two positive integers $n,k$.

For all $\forall i \in [0,k)$, when generating a uniformly random permutation of length $n$, compute the probability that the number of inversions in the permutation modulo $k$ has remainder $i$. Output the answers modulo $998244353$.

Input Format

One line containing two integers $n,k$.

Output Format

One line containing $k$ integers separated by spaces. The $i$-th integer represents the probability that the number of inversions in the permutation modulo $k$ has remainder $i-1$.

4 5

166374059 166374059 457528662 748683265 457528662

Hint

Sample Explanation

Constraints

For $100\%$ of the testdata, it is guaranteed that $1\le n\le 10^5$, $2\le k\le 1000$.

Re-collected from XDUCPC 2021 Onsite Contest F.

Translated by ChatGPT 5