CF1687F.Koishi's Unconscious Permutation

As she closed the Satori's eye that could read minds, Koishi gained the ability to live in unconsciousness. Even she herself does not know what she is up to.

— Subterranean Animism

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 $998244353$ .

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 $f(p)=1$ .