1307.2010 (J. Fernando Barbero G. et al.)

Bivariate Generating Functions for a Class of Linear Recurrences. I.
General Structure    [PDF]

J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor

We consider the Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashkin, and solve it using bivariate exponential generating functions. This problem includes many particular cases of great combinatorial interest. We find a complete classification in four types of the solution for this problem, and for each type, we obtain the corresponding exponential generating function. For some families the generating function requires the introduction of a generalization of the tree function. We provide many applications of our general results to particularly interesting cases.

View original: http://arxiv.org/abs/1307.2010