I believe that there is a statement along the following lines (I would, of course, love to be corrected): a formal power series is the Taylor expansion of a rational function if and only if the coefficients eventually satisfy a linear relationship.

Let's suppose that I understand what "satisfy a linear relationship" means, because it's not the part I actually want to ask about (although clarifications are very welcome!). What I would like to know is what conditions on a rational function are equivalent to all the Taylor coefficients being nonnegative integers. For example, I happen to know that $1/(1-kx) = \sum (kx)^n$, and so any sum or product of such functions works. In particular, I can try playing around with partial-fraction decompositions to see if I can write a given rational function in this way. But I have no idea if this is all of them.

Put another way, there is a map $\mathbb R(x) \to \mathbb R[x^{-1},x]]$ (rational functions to Laurent series). I would like to understand the inverse image of $\mathbb N[x^{-1},x]]$.

(Oh, also, I have no idea how to tag this, and I think "general mathematics" is probably an inappropriate tag for MO. So please re-tag as you see fit.)

quasi-polynomial-- see Section 3 of cs.uwaterloo.ca/journals/JIS/VOL8/Clark/clark80.pdf Note that this paper is concerned with a family of rational functions whose Taylor series have non-negative coefficients, so this might be of interest to you. $\endgroup$