An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square_n$ denote all such paths. Define the area of a path $P$ to be the number of full squares below the path and above the diagonal. Then one variant of a $q$-Catalan number is given by $$C_n(q)=\sum_{P\in\square_n}q^{area(P)}.$$ A few examples: $C_1(q)=1,\, C_2(q)=1+q,\, C_3(q)=1+2q+q^2+q^3$.

There is this notion of $q,t$-Catalan numbers so that $C_n(q)=C_n(q,1)$.

I'm finding (after some fooling around) the following generating function which I've not encountered in the literature.

Question. Is this known or can you provide a proof? $$\sum_{n=0}^{\infty}C_n(q)\,z^n =\frac{\sum_{k=0}^{\infty}q^{k^2}\prod_{j=1}^k\frac{z}{q^j-1}} {{\sum_{k=0}^{\infty}q^{k(k-1)}\prod_{j=1}^k\frac{z}{q^j-1}}}$$