F. Koishi's Unconscious Permutation
time limit per test
12 seconds
memory limit per test
1024 megabytes
input
standard input
output
standard output
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 $$$998244353$$$.

Examples
Input
2 0
Output
1 0
Input
4 1
Output
0 3 6 0
Input
8 3
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 $$$f(p)=1$$$.