Look into generatingfunctionology by Herbert S. Wilf.
It's precisely about generating functions! Fairly short but thorough. From the link (Wilf's website), you can
- download the $2$nd edition of the text (as a pdf document),
- download a version of the book with hyperlinks to be read on-line,
- or else click on the link to the publisher if you'd like to purchase the $3$rd edition.
- (Perhaps you can obtain a hard copy from you library.)
From MIT (and a course taught there), you can download a pdf dedicated to generating functions, which is outlined nicely, and allows you to access the topics in order, or those of interest.
Finally, see Ch. 10: Generating Functions of the freely available text authored by Grinstead and Laurie Snell, and published by the AMS: Introduction to Probability.
You might want to invest in Concrete Mathematics by Graham, Knuth, and Patashnik. It has a nicely written introduction to generating functions,