I would like to know if there is an explicit formula for the number of different (non isomorphic) simple graphs with a given number of vertices $p$ and edges $q$, and if yes what is it.

Trying to find it I've stumbled on an earlier question: Counting non isomorphic graphs with prescribed number of edges and vertices which was answered by Tony Huynh and in this answer an explicit formula is mentioned and said that it can be found here, but I can't find it there so I need help. Basically what I'm looking for is an explicit formula for $g_{p q}$ (of the form $g_{pq} = \; ...$ ) which is mentioned in the given link (and, for example, the book "Graphical Enumeration" by Harary and Palmer on page 82), or a way to obtain it.

The reason I need this is because I'm doing research on the statistical "physics" of (complex) networks and I want to see what I get when I define entropy using this... Thanks to all in advance :)