A15026. Centroid Probabilities

Consider every tree (connected undirected acyclic graph) with $n$ vertices ( $n$ is odd, vertices numbered from $1$ to $n$ ), and for each $2 \le i \le n$ the $i$ -th vertex is adjacent to exactly one vertex with a smaller index.

For each $i$ ( $1 \le i \le n$ ) calculate the number of trees for which the $i$ -th vertex will be the centroid. The answer can be huge, output it modulo $998\,244\,353$ .

A vertex is called a centroid if its removal splits the tree into subtrees with at most $(n-1)/2$ vertices each.

The first line contains an odd integer $n$ ( $3 \le n < 2 \cdot 10^5$ , $n$ is odd) — the number of the vertices in the tree.

Print $n$ integers in a single line, the $i$ -th integer is the answer for the $i$ -th vertex (modulo $998\,244\,353$ ).

Example $1$ : there are two possible trees: with edges $(1-2)$ , and $(1-3)$ — here the centroid is $1$ ; and with edges $(1-2)$ , and $(2-3)$ — here the centroid is $2$ . So the answer is $1, 1, 0$ .

Example $2$ : there are $24$ possible trees, for example with edges $(1-2)$ , $(2-3)$ , $(3-4)$ , and $(4-5)$ . Here the centroid is $3$ .