P7857 "EZEC-9" Meltel

> Only you are unforgivable. > "Only you cannot be forgiven." > Only you forgave everything about me. > "Only you forgave all of me." > Only you took everything from me, devoured it all, > "Only you destroyed my world," > burned it all to ashes and turned everything into nothing. > "Burned it to the ground, mountains collapsing and the earth splitting."

Given a positive integer $n$, for $s=0,1,\dots,n$, count the number of labeled forests consisting of labeled rooted trees on $n$ nodes such that there exist exactly $s$ binary trees. Take the result modulo $998244353$. A binary tree is defined as a labeled rooted tree where each node has at most $2$ children. Two forests are considered different if and only if there exists a node whose parent is different (treat the root as having no parent).

The first line contains one positive integer $n$.

Output one line with $n+1$ non-negative integers, representing the answers for $s=0,1,\dots,n$.

[Explanation for Sample 1] With $3$ nodes, only binary trees can appear, so the number of valid configurations for $s=0$ is $0$. There are $9$ labeled rooted binary trees on $3$ nodes, so the number of valid configurations for $s=1$ is $9$. There are $2$ labeled rooted binary trees on $2$ nodes, so the number of valid configurations for $s=2$ is $3 \times 2 = 6$. A single node is also considered a labeled rooted binary tree, so the number of valid configurations for $s=3$ is $1$. [Constraints] **This problem uses bundled testdata.** - Subtask 1 (5 points): $n \le 5$. - Subtask 2 (5 points): $n \le 10$. - Subtask 3 (30 points): $n \le 10^3$. - Subtask 4 (30 points): $n \le 8 \times 10^3$. - Subtask 5 (30 points): no special constraints. For $100\%$ of the testdata, $1 \le n \le 2 \times 10^5$。 Translated by ChatGPT 5