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 $998\,244\,353$ ).