suppose, for the sake of me practicing writing mathml, that you have a generating function taking the form of a rational function, whose coefficients you'd like to extract.
suppose further that you have to do this
by hand 
instead of trying to futz around with (because you know already that the iterated derivatives of are going to be so annoying) there's a neat trick you can do involving the geometric series formula. wlog check this shit out:
where of course and and are easily determined from and .
imo this simple product form is good actually for generating function–style thinking. it even has a reasonably simple interpretation in terms of ℕ-weighted sets and the kleene star: whatever is counting, it's basically something that (up to a shift of ) decomposes into a and a (i.e. a finite string of s).
unfortunately you may need some mental gymnastics à la virtual species (though of course we have ordinary genfuncs here and not EGFs) to interpret negative terms in and especially . but i think this is a relatively small price to pay for algebra this clean