Given a rational ordinary generating function we can directly compute the coefficients of that generating function.
What is the broadest set of generating functions that allow for direct computation of their coefficients? I'm interested in both ordinary and exponential generating functions.
With direct computation I mean that there is an algorithm that directly recognize one of multiple forms and apply the appropriate computation to find the coefficients, not a generic computer algebra system that can figure out the right answer for a broad but unclear set of generating functions.