#P1687F. Koishi's Unconscious Permutation
Koishi's Unconscious Permutation
Description
Koishi is unconsciously permuting $n$ numbers: $1, 2, \ldots, n$.
She thinks the permutation $p$ is beautiful if $s=\sum\limits_{i=1}^{n-1} [p_i+1=p_{i+1}]$. $[x]$ equals to $1$ if $x$ holds, or $0$ otherwise.
For each $k\in[0,n-1]$, she wants to know the number of beautiful permutations of length $n$ satisfying $k=\sum\limits_{i=1}^{n-1}[p_i<p_{i+1}]$.
There is one line containing two intergers $n$ ($1 \leq n \leq 250\,000$) and $s$ ($0 \leq s < n$).
Print one line with $n$ intergers. The $i$-th integers represents the answer of $k=i-1$, modulo $998244353$.
Input
There is one line containing two intergers $n$ ($1 \leq n \leq 250\,000$) and $s$ ($0 \leq s < n$).
Output
Print one line with $n$ intergers. The $i$-th integers represents the answer of $k=i-1$, modulo $998244353$.
Samples
2 0
1 0
4 1
0 3 6 0
8 3
0 0 0 35 770 980 70 0
Note
Let $f(p)=\sum\limits_{i=1}^{n-1}[p_i<p_{i+1}]$.
Testcase 1:
$[2,1]$ is the only beautiful permutation. And $f([2,1])=0$.
Testcase 2:
Beautiful permutations:
$[1,2,4,3]$, $[1,3,4,2]$, $[1,4,2,3]$, $[2,1,3,4]$, $[2,3,1,4]$, $[3,1,2,4]$, $[3,4,2,1]$, $[4,2,3,1]$, $[4,3,1,2]$. The first six of them satisfy $f(p)=2$, while others satisfy $f(p)=1$.