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:

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