A13387. Slime and Sequences (Hard Version)

Note that the only differences between easy and hard versions are the constraints on $n$ and the time limit. You can make hacks only if all versions are solved.

Slime is interested in sequences. He defined good positive integer sequences $p$ of length $n$ as follows:

- For each $k>1$ that presents in $p$ , there should be at least one pair of indices $i,j$ , such that $1 \leq i < j \leq n$ , $p_i = k - 1$ and $p_j = k$ .

For the given integer $n$ , the set of all good sequences of length $n$ is $s_n$ . For the fixed integer $k$ and the sequence $p$ , let $f_p(k)$ be the number of times that $k$ appears in $p$ . For each $k$ from $1$ to $n$ , Slime wants to know the following value:

$\left(\sum_{p\in s_n} f_p(k)\right)\ \textrm{mod}\ 998\,244\,353$

The first line contains one integer $n\ (1\le n\le 100\,000)$ .

Print $n$ integers, the $i$ -th of them should be equal to $\left(\sum_{p\in s_n} f_p(i)\right)\ \textrm{mod}\ 998\,244\,353$ .

In the first example, $s=\{[1,1],[1,2]\}$ .

In the second example, $s=\{[1,1,1],[1,1,2],[1,2,1],[1,2,2],[2,1,2],[1,2,3]\}$ .

In the third example, $s=\{[1]\}$ .