P16722 Basic Modular Form Practice Problem.

~~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?

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 $998244353$.

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

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

**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