I don’t have much to say here apart from a couple of links and references I need from time to time.
But nor should I; simply download Herbert Wilf’s pellucid free textbook for all the possible introduction you could need.
Rolf Bardeli walks through some beautiful generating function application.
Later I would like to include notes on graph theory and neat count RV tricks based on generating functions. You an rapidly get into weird complex analysis when you consider asymptotics of these guys and get to find out why Cauchy’s integral formula and Lagrange’s Inversion Theorem were in your textbook.
Consul, P. C., and L. R. Shenton. 1973. “Some Interesting Properties of Lagrangian Distributions.” Communications in Statistics 2 (3): 263–72. https://doi.org/10.1080/03610927308827073.
Consul, P., and L. Shenton. 1972. “Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions.” SIAM Journal on Applied Mathematics 23 (2): 239–48. https://doi.org/10.1137/0123026.
Janardan, K. 1984. “Moments of Certain Series Distributions and Their Applications.” SIAM Journal on Applied Mathematics 44 (4): 854–68. https://doi.org/10.1137/0144061.
Mutafchiev, Ljuben. 1995. “Local Limit Approximations for Lagrangian Distributions.” Aequationes Mathematicae 49 (1): 57–85. https://doi.org/10.1007/BF01827929.
Saichev, A., and D. Sornette. 2011. “Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes,” January. http://arxiv.org/abs/1101.5564.
Wilf, Herbert S. 1994. Generatingfunctionology. Boston: Academic Press. http://catalog.hathitrust.org/api/volumes/oclc/29831213.html.