How to determine growth rate of coefficients of generating function

Question

For a given ordinary generating function $f(x)=a_0+a_1x+...$, are there any methods to determine the growth rate of its coefficients based on that of $f$ ? In particular if we are given the extra information that $$\lim\limits_{a \to 1^-}\frac{1-a}{a}*\log f(a)=\frac{\pi^2}{12}$$ Can we strengthen our asymptotic estimate?

Answer 1

Determining the asymptotic growth of the coefficients of a generating function is essentially the entire subject of Analytic Combinatorics, on which see the 810-page book by Flajolet and Sedgewick available online.

The basic setup is this: for a large class of generating functions $G(z) = g_0 + g_1z + g_2z^2 + \dots$, we can show that the coefficients satisfy $$g_n = [z^n]G(z) = A^n \theta(n),$$ where $A^n$ is the exponential rate of growth and $\theta(n)$ is a subexponential factor.

Here, the exponential rate of growth $A = \lim \sup |g_n|^{1/n}$ is determined by the location of the singularities of the function $G(z)$: under nice conditions, it is simply the reciprocal of the radius of convergence ($A = 1/R$).

When $A > 1$, for many purposes it's sufficient to determine $A$, but when $A = 1$ it's not very informative.

The subexponential factor $\theta(n)$ is determined by the "nature" of the singularities (in a way that's made clear in the book).

Answer 2

By the Cauchy-Hadamard theorem, $\limsup\sqrt[n]{|a_n|}=1/R$. If the lim sup is in reality a lim, and even better, if the limit of the quotient formula exists, then the asymptotic growth rate is about the reciprocal of the distance of the origin to the closest singularity of $f$.