TITLE: How can I learn about generating functions?
QUESTION [30 upvotes]: The intent of this question is to provide a list of learning resources for people who are new to generating functions and would like to learn about them.  
I'm personally interested in combinatorics, and I sometimes use generating functions in answers to combinatorial questions on stackexchange, but I know many readers are not familiar with these objects.  I hope this list will help those readers and provoke interest in GFs generally. 
I will provide an answer below, but feel free to edit my answer or provide your own answer.  Initially it will be a short list, but maybe it will grow over time.  Please regard this question and its answers as a community resource.
Acknowledgement: In asking this self-answered question I was prompted by helpful advice provided by this discussion on mathematics meta stackexchange, users quid and John Omielan in particular.  Thank you.

REPLY [2 votes]: One book which is worth mentioning is Irresistible Integrals by George Boros and Victor Moll. It touches a bit on GFs, especially in relation to their use in the evaluation of series and integrals, as well as their connections to the study of polynomials and recurrence relations. 
One of the first chapters uses the recursive definition of the Fibonacci numbers in order to find their generating function, namely $$\sum_{n\geq0}F_nx^n=\frac1{1-x-x^2}$$
But the use of GF's is consistent throughout the book. I highly recommend it if you also want to learn about series, integrals, and polynomials.