I finished my first discrete math course this semester using mostly the excellent Kenneth Rosen (Discrete Mathematics and Applications) book that was a great help, especially in induction content and recurrence relations. In the meantime, I have read some criticisms of the work on Amazon and would like recommendations that would close some of the gaps that the author has opened, especially in the areas of integer partitions and generating functions, and which allow for the deepening of themes such as mathematical logic.
The book Discrete Math by Gary Chartrand and Ping Zhang would likely be an excellent resource. The ISBN-13 is 978-1577667308.
The book opens with a number of chapters on proofs in mathematics using logic and a variety of methods of proof. The book also goes into some good detail on combinatorics. I know that generating functions are included in Chapter 9.