Given a meromorphic function $f(z)$ and a point $z_0$ which isn't a pole, we can compute the Taylor expansion of $f$ about $z_0$, i.e., $f(z)=\sum a_n (z-z_0)^n$ where $a_n=f^{(n)}(z_0)/n!$. If one wishes to know all the coefficients, then one can either find a pattern in the derivatives of $f$ or in the contour integral representation of the coefficients. However, it is possible to say something about the coefficients without explicitly computing them all. For example, the radius of convergence of the Taylor series, which can be used as a measure of the rate of growth of the coefficients, is equal to the distance from $z_0$ to the nearest pole of $f$. Similarly, if $f(z_0+x)$ is real whenever $x$ is real, then we can say that coefficients are all real (I think?)
Are there other interesting properties about the coefficients that can be determined without explicitly computing them? For example, is there an easy test to determine of the coefficients are rational? Or if either $a_n$ or $n!a_n$ are integers? Or if we know the coefficients are real, can we determine if they are positive? Is there a simple condition that implies the coefficients are bounded? More generally, what properties of a sequence are reflected in properties of its generating function, and vice versa. Is there a book where questions like this are discussed?
N.B. This is not for any particular purpose, I was just curious, and have not had much luck with google. I realize the question is a bit vague, so if it is better as community-wiki, I humbly ask a mod to mark it as such.