Koishi's Unconscious Permutation

12000ms

1048576K

Description:

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 $$$k=\sum\limits_{i=1}^{n-1}[p_i<p_{i+1}]$$$.

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 $$$998244353$$$.

Sample Input:

2 0

Sample Output:

1 0

Sample Input:

4 1

Sample Output:

0 3 6 0

Sample Input:

8 3

Sample Output:

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 $$$f(p)=1$$$.

Informação

Provedor Codeforces

Código CF1687F