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.

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.

Edit : Doing the computations I find, \begin{align*} Bip(xt,yt) &= 1+2xy{t}^{2}+ \left( 2y{x}^{2}+2{y}^{2}x \right) {t}^{3}+ \left( 5{y}^{2}{x}^{2}+2y{x}^{3}+2{y}^{3}x \right) {t}^{4} + ... \end{align*} which is wrong. For example we should have $x^2 + y^2 +2xy$ in front of $t^2$, a $xt$ term as well as a $yt$ term..