codeforces#P1667E. Centroid Probabilities
Centroid Probabilities
Description
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 $998\,244\,353$).
Input
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.
Output
Print $n$ integers in a single line, the $i$-th integer is the answer for the $i$-th vertex (modulo $998\,244\,353$).
Samples
3
1 1 0
5
10 10 4 0 0
7
276 276 132 36 0 0 0
Note
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 $3$.