Background

As everyone knows the following identities hold for any positive integer $n$:

$$12\sum_{i+j=n} \sigma_1(i)\sigma_1(j)=5\sigma_3(n)-(6n-1)\sigma_1(n)$$$$192 \sum_{i+j+k=n}\sigma_1(i)\sigma_1(j) \sigma_1(k)=7\sigma_5(n)+(10-30n)\sigma_3(n)+(1-12n+24n^2)\sigma_1(n)$$

How can this property be generalized to the general case?

Problem Description

Let

$$f(x)=1-24\sum_{i \geq 1} \frac{i x^i}{1-x^i}$$

Given positive integers $p, k, n$, where $p$ is guaranteed to be a prime, find the coefficient of $x^{p^n}$ in $f(x)^k$.
The answer may be very large; you only need to output it modulo $998244353$.

Input Format

One line containing three positive integers $p, k, n$.

Output Format

Output one integer on one line, representing the answer modulo $998244353$.

2 4 3

258593760

5 6 6

177906112

3 18 8

566697133

2 24 17

401935277

2 96 18

36903708

Hint

This problem uses bundled testdata.

Subtask 1 (10 pts): $1 \le p^n \le 10^6$;
Subtask 2 (20 pts): $1 \le k \le 6$;
Subtask 3 (30 pts): $1 \le p^{k/6} \le 10^6$;
Subtask 4 (40 pts): No special constraints.

For all testdata: $1 \le k \le 96$, $1 \le n \le 10^9$, $p \in \{ 2, 3, 5, 7 \}$.

Translated by ChatGPT 5