I have attempted to calculate the number of unlabelled bipartite graphs as follows:

Let $G = (V_1, V_2, E)$ be a bipartite graph on $n$ vertices with $|V_1| = m$ and $|V_2| = n-m$. Assume without loss of generality that $|V_1| \leq |V_2|$ so $m \leq \left\lfloor \frac{n}{2} \right\rfloor$. If $G$ is complete bipartite then it has $m(n-m)$ edges since each of the vertices in $V_1$ is connected to each in $V_2$. Thus, the total number of bipartite graphs with parts of size $m$ and $n-m$ is $2^{m(n-m)}$. In order to find the total number of possible bipartite graphs on $n$ vertices we sum over all possible $m$: \begin{align} \sum^{\left\lfloor \frac{n}{2} \right\rfloor}_{m=1} 2^{m(n-m)} \end{align}

However, I notice that I have counted labelled bipartite graphs where I need the number of unlabelled graphs. I'm struggling to see how to account for this.