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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.

Here are some resources to get you started on generating functions.

• 
  generatingfunctionology by Herbert S. Wilf is probably the best introductory text. You can find this book in pdf format for free, online, but I think it's worth adding a hard copy to your library. The writing style is breezy and entertaining. First sentence: "A generating function is a clothesline on which we hang up a sequence of numbers for display."



• An Introduction to the Analysis of Algorithms by Robert Sedgewick and Philippe Flajolet is another fine introduction. Despite its title, the book is mostly about generating functions. Coursera has a free online course from Princeton University based on this book, with Professor Sedgewick as the instructor,so here is your chance to sit at the feet of the Master, figuratively: Coursera Analysis of Algorithms. There is also a follow-on course on Analytic Combinatorics: Coursera Analytic Combinatorics. The analytic combinatorics course is based on the book Analytic Combinatorics by the same two authors, which is a fine book, but I don't think it's the best starting point for most beginners. (Of course, you might be the exception, and it really is a wonderful book with encyclopedic coverage.)




• While we are on the subject of free online materials, Bogart's Introductory Combinatorics (pdf) includes an introduction to generating functions.




• Schaum's Outline of Theory and Problems of Combinatorics by V.K. Balakrishnan includes a good introduction to generating functions and is noteworthy for being inexpensive by comparison with other texts (around $20 on Amazon the last time I checked). I find this book useful both for reference and as a learning resource. Coverage includes some relatively advanced topics, such as rook polynomials and the Polya Enumeration Theorem.




• And there are many others, I am sure. Many books on combinatorics include sections on generating functions.

As for prerequisites, many applications of GFs require only a knowledge of polynomials. But many require infinite series, so you need some exposure to series like infinite geometric series, the series for $e^x$, and the Binomial Theorem for negative and fractional exponents. Interestingly enough, we can often (but not always) ignore questions of convergence, because we view the series as formal objects and don't worry about evaluating them. Some applications use differential equations or complex analysis, but you can go a long way without these.

I would like to end with a little story, in hope of getting more people interested in GFs. Here is how the statistician Frederick Mosteller described his initial exposure to generating functions.

A key moment in my life occurred in one of those classes during my
sophomore year. We had the question: When three dice are rolled what
is the chance that the sum of the faces will be 10? The students in
this course were very good, but we all got the answer largely by
counting on our fingers. When we came to class, I said to the teacher,
"That's all very well - we got the answer - but if we had been asked
about six dice and the probability of getting 18, we would still be
home counting. How do you do problems like that?" He said, "I don't
know, but I know a man who probably does and I'll ask him." One day I
was in the library and Professor Edwin G Olds of the Mathematics
Department came in. He shouted at me, "I hear you're interested in the
three dice problem." He had a huge voice, and you know how libraries
are. I was embarrassed. "Well, come and see me," he said, and I'll
show you about it." "Sure, " I said. But I was saying to myself, "I'll
never go." Then he said, "What are you doing?" I showed him. "That's
nothing important," he said. "Let's go now."

So we went to his office, and he showed me a generating function. It
was the most marvelous thing I had ever seen in mathematics. It used
mathematics that, up to that time, in my heart of hearts, I had
thought was something that mathematicians just did to create homework
problems for innocent students in high school and college. I don't
know where I had got ideas like that about various parts of
mathematics. Anyway, I was stunned when I saw how Olds used this
mathematics that I hadn't believed in. He used it in such an unusually
outrageous way. It was a total retranslation of the meaning of the
numbers. [Albers, More Mathematical People].

One of the treasures which might fit the needs is Concrete Mathematics
by R.L. Graham, D.E. Knuth and O. Patashnik.

A starting point could be section 5.4 Generating Functions where we can read:

• We come now to the most important idea in this whole book, the
  notion of a generating function. ...

The book provides a wealth of instructive examples devoting chapter 7 Generating Functions exclusively to the subject of interest.