In the formalism of species of structure. The species $Bip(X,Y)$ of bipartite graphs on two sorts of vertex $X$ and $Y$ can be described as,
$$
Bip(X,Y) \simeq E^2 \circ (E^\bullet(X) E^\bullet(Y)) \simeq E^2 \circ (XY E(X + Y))
$$
where $E$ is the species of sets, $\circ$ is the functorial composition of species and $\bullet$ is the pointing operator.

If I make no mistake the series you search is,
$$
Bip(x,y) = \prod_{k \ge 1}\exp\left[\frac{2}{k}x^ky^k\prod_{l \ge 1} \exp\left(\frac{1}{l}(x^{kl}+y^{kl}) \right)\right]
$$

Hope this helps.