My question can be stated as follows: let $X$ be a hereditary family of unlabelled graphs closed under disjoint unions. Suppose we know, for each $n$, the number $c_n$ of connected graphs in X on $n$ vertices. How do we get from that the total number $x_n$ of graphs in $X$ on $n$ vertices?
I am vaguely aware that there are methods involving generating functions that answer (some version of) this question, but the only source I found for that online is a transcription of a set of lecture notes with no references whatsoever. I am not particularly well versed in the theory of generating functions, so any source which goes from the basics to the relevant derivation would be greatly appreciated.
However, if it is at all possible, I would ideally like to find an explicit formula for the coefficients $x_n$ in terms of the $c_i$, or at least a (tight-ish) lower bound for $x_n$ in terms of the $c_i$.
I can see the answer in some very simple cases: if, say, $c_i = 1$ for all $i$ (e.g., $X$ is the class of unions of cliques), then $x_n$ is just the partition function $p(n)$ from number theory (compare this to the labelled case, where we get the Bell numbers). However, I do not see how to generalise this. It seems to me very likely that this question would have been studied in the language of multisets, but once again, I am not particularly familiar with that area.