I have computed a generating function for a problem involving a particular series, and would like to know if anyone has any references or a categorisation for it? It's $$ G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}. $$ It appears to be related to (mock) theta functions, but seems to be simpler. In particular, I would like to know whether $G(a,z)$ satisfies any identities?

Many thanks.