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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 $$$998\,244\,353$$$).
3
1 1 0
5
10 10 4 0 0
7
276 276 132 36 0 0 0
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 $$$3$$$.
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Provedor Codeforces
Código CF1667E
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Datas 09/05/2023 10:27:15
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