Borwein, J. M.; Crandall, R. E.: Closed Forms: What They Are and Why We Care. Notices of the AMS 60 (2013) (1) 50-65
Shackell, J. R.: Symbolic ...
I am fairly certain this is false in general (although it works for certain special $S$ and might work for small values of $d$, but when $d$ is sufficiently large relative to $k=|V(S)|$ it shouldn't work). Here's why, and some reasoning that might help you characterize exactly for which $S$ it works.
Fix $S$, and let $s(G)$ denote the number of copies of $S$...
I apologize in advance for my abuse of set notation as I am mapping one mathematical object that is not quite a set i.e. graphs to another mathematical object that is not quite a set i.e. sequences. Feel free to comment or correct notation etc.
(1) The triangular-harmonic numbers $\Theta$ are a monotonic sequence of rational numbers greater ...