Fridgeberg is a well-known veteran in the Taiwanese ICPC community. He specializes in counting problems. Since he solves counting problems faster than everyone else, people call him a speed-counter. Today, he has prepared a challenge for you along with the following message.
"Yo, I'm Fridgeberg, and I'm an 18-year-old ICPC speed-counter from Yunlin, Taiwan. Today, I'm challenging you to a counting problem."
The challenge is as follows.
A rooted labelled tree with $$$n$$$ vertices is a tree with vertices labelled $$$1,2,\ldots,n$$$ and one vertex chosen as the root.
Recall that for any two vertices in a tree, there is a unique simple path connecting them. For each vertex $$$v$$$ in a rooted labelled tree, define its subtree as the set of all vertices $$$u$$$ such that $$$v$$$ is on the unique path from the root to $$$u$$$. In particular, a vertex is always in its own subtree. The subtrees of a tree are the collection of all vertices' subtrees.
A rooted labelled tree is $$$k$$$-free if none of its subtrees has exactly $$$k$$$ vertices.
For each $$$k$$$ from $$$2$$$ to $$$n-1$$$, compute the number of different rooted labelled trees with $$$n$$$ vertices that are $$$k$$$-free. Since the numbers can be very large, please compute them modulo $$$998244353$$$.
Two rooted labelled trees are different if at least one of the following holds:
The only line contains a single integer $$$n$$$.
Output $$$n-2$$$ lines. On the $$$i$$$-th line, output the answer when $$$k=i+1$$$.
3
3
4
16 28
11
23790225 701354623 477759311 29734262 735403659 26527433 622702793 495236333 962003659
Explanation for Sample 1
There are exactly $$$3$$$ rooted labelled trees that have no subtree with exactly $$$2$$$ vertices.
The following picture illustrates those $$$3$$$ trees. (The root vertices are colored pink, and the other vertices are colored light blue.)
