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Sorry for a possibly off-the-topic question, but I am afraid to gain the necessary overview to give an answer (supposed the question is not ill-posed) is beyond my capabilities. In the course of ...

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 $n$ ...