#P1916H2. Matrix Rank (Hard Version)
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ID: 10278
Type: RemoteJudge
2000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: 10
Uploaded By:
Hydro
Tags> Matrix Rank (Hard Version)
Description
This is the hard version of the problem. The only differences between the two versions of this problem are the constraints on $k$. You can make hacks only if all versions of the problem are solved.
You are given integers $n$, $p$ and $k$. $p$ is guaranteed to be a prime number.
For each $r$ from $0$ to $k$, find the number of $n \times n$ matrices $A$ of the field$^\dagger$ of integers modulo $p$ such that the rank$^\ddagger$ of $A$ is exactly $r$. Since these values are big, you are only required to output them modulo $998\,244\,353$.
$^\dagger$ https://en.wikipedia.org/wiki/Field_(mathematics)
$^\ddagger$ https://en.wikipedia.org/wiki/Rank_(linear_algebra)
The first line of input contains three integers $n$, $p$ and $k$ ($1 \leq n \leq 10^{18}$, $2 \leq p < 998\,244\,353$, $0 \leq k \leq 5 \cdot 10^5$).
It is guaranteed that $p$ is a prime number.
Output $k+1$ integers, the answers for each $r$ from $0$ to $k$.
Input
The first line of input contains three integers $n$, $p$ and $k$ ($1 \leq n \leq 10^{18}$, $2 \leq p < 998\,244\,353$, $0 \leq k \leq 5 \cdot 10^5$).
It is guaranteed that $p$ is a prime number.
Output
Output $k+1$ integers, the answers for each $r$ from $0$ to $k$.
3 2 3
1 51549919 2
1 49 294 168
1 51549918 0