Asymptotics of Generating Functions

Given a generating function $A(x)$, are there any general techniques for finding the asymptotics of the associated sequence? For example, given the generating function satisfying $A(x) = 1 + x\cdot A\left(\frac{x}{A(-x)}\right)$, is there anything we can say about how fast the coefficients grow?

• In your example, you are not "given" a generating function, you are given a functional equation for a generating function. Not at all the same thing. Aug 3, 2017 at 2:25
• BTW, your sequence is OEIS sequence A211768. If you do find asymptotics for it, please contribute that to the OEIS. Aug 3, 2017 at 2:32
• You are correct, I meant a generating function satisfying a specific equation such as the one above and will change the question accordingly. Given this change in the nature of the question, is there anything that can be deduced about the asymptotics of the sequence from such a functional equation? Aug 3, 2017 at 2:53
• The OEIS sequence is actually why I am interested in this question, since this generating function is the only way the terms of the sequence are specified Aug 3, 2017 at 2:55