What characterizes rational functions with nonnegative integer Taylor coefficients? 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.)
 A: It is apparently «unknown whether the problem "$a_n > 0$ for all $n$?" is decidable for linear recurrence sequences», according to these notes by Stefan Gerhold.
A: By linear relationship, you mean a linear recurrence relation. Do it by writing down $$ P(z)/Q(z) = \sum_n a_nx^n $$, multiply by $Q(z)$ on both sides, regroup terms. You'll get something like $$ 0 = \sum_n (c_1a_{n+k} + c_2 a_{n+k-1} + \dots + c_ka_n)x^n $$ so the coefficients do indeed satisfy a linear recurrence.
Furthermore, I think that a function, with radius of convergence 1, and positive integer coefficients, has to be either a rational function (in which case the coefficients satisfy a linear recurrence relation) or has natural boundary on the circle $|z| = 1$ (in which case the function could be taught off as rather complicated).
So in the end, your question is really a question about linear recurrences relations, rather than about rational functions. I think Narkiewicz did some work in the field, but I have no reference on top of my head.
A: This paper (?) of Gessel might help you out, although it is mostly about combinatorics.  There are two natural ways to write down rational functions with non-negative integer coefficients in combinatorics, one coming from transfer matrices / finite automata and one coming from regular languages.  The two give the same class of rational functions, but there exist rational functions with non-negative integer coefficients which provably don't arise in this way, so the situation seems complicated.
Your question seems to indicate you're not familiar with this class of rational functions, so here are two equivalent definitions: it is, on the one hand, the class of all non-negative linear combinations of entries of matrices of the form $(\mathbf{I} - \mathbf{A})^{-1}$ where $\mathbf{A}$ is a square matrix with entries in $x \mathbb{N}[x]$, and on the other hand the minimal class of rational functions containing $1, x$, and closed under addition, multiplication, and the operation $f \mapsto \frac{1}{1 - xf}$.
Edit:  One reason there isn't likely to be a particularly nice classification is that one can start with any rational function with integer coefficients and add a polynomial and a term of the form $\frac{1}{1 - kx}$ for $k$ such that $\frac{1}{k}$ is smaller than the smallest pole.  
