Number of partitions of $n$ in which parts can occur at most 7 times

Question

I have got the following answer without detailing the whole answer as I am looking precisely for the notation used here as I am new to graph theory, so please pardon me. After a series of operations, I got the following

$$A(x)=\sum\limits_{n=1}^{\infty} p(n)q^n = ... = \frac{1}{(9^6\times9^6)_{\infty}(9^6\times9^6)_{\infty} (9^6\times9^2)_{\infty}(9^4\times9^6)_{\infty}}$$

What does $\infty$ notation here referring to?

Answer 1

It looks like a bad OCR scan. I suspect that some of the $9$s are supposed to be $q$s, and you have mixed up $x$ and $q$.

If $p(n)$ is the number of such partitions, the correct generating function is $$A(x) = \sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty \sum_{i=0}^7 x^{ki} = \prod_{k=1}^\infty \frac{1-x^{8k}}{1-x^k}$$