Is there a a simple expression for the following limit,
$$t(a, q) = \lim_{n\rightarrow \infty} \left(-1\right)^{n}\frac{a^n q^{\frac{n^2-n}{2}}}{\left(a;q\right)_n} .$$
The denominator $\left(a; q\right)_n$ is the $q$-Pochhammer symbol, and the numerator corresponds to the norm of the $n$'th coefficient of the expansion of $\left(a;q\right)_n$. The question arised from investigating the asymptotic behaviour of a combinatorial formula, and $t(a, q)$ is the scaling factor. I am particularly interested in the case $a=q$.