Let
\begin{equation}
G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)
\end{equation}
be a bigraded set of generators and $\mathcal{F}(G)$ be the *free PRO generated by $G$* (see [1] for a net description of free PROs).

Let us denote by $H_G(x, y, z)$ the *generating series of $\mathcal{F}(G)$*, defined by
\begin{equation}
H_G(x, y, z) :=
\sum_{p \geq 0} \: \sum_{q \geq 0} \: \sum_{e \in \mathcal{F}(G)(p, q)} x^p y^q z^{\deg e},
\end{equation}
where $\deg e$ is the *degree* of $e \in \mathcal{F}(G)$, that is the number of generators of $G$ needed to build $e$. In other words, $H_G(x, y, z)$ is the generating series that counts the elements of $\mathcal{F}(G)$ by the input arity (parameter $x$), by the output arity (parameter $y$), and by the degree (parameter $z$).

My question is the following:

Given a set $G$ of generators, do we know an expression for the generating series $H_G$ for $\mathcal{F}(G)$?

I think, since this question is very basic and natural, that it would appear in some papers dealing with PROs. References are thus welcome.

[1] Y. Lafont, Diagram rewriting and operads, Operads 2009, ed. J.-L. Loday & B. Vallette, SÃ©minaires et CongrÃ¨s 26, p. 163-179, SMF, 2011.