Doubly exponential generating function

Question

Miklos Bona, Introduction to Enumerative Combinatorics, Rudin, 2007, Section 3.5 , page 173 gives the definition and one example of a doubly exponential generating function. The author states in the introduction to this section that these are used "in one of the later chapters". Does anyone know exactly where else in this book these functions are discussed. Does anyone have any other references that I could find on-line to give myself an introduction to this type of generating function.

Answer 1

The particular example in Bona is the OEIS sequence A061573. See OEIS sequence A000275 for another example which is $$ 1/J_0(\sqrt{4x}) = \sum_{n=0}^\infty a_n \frac{x^n}{n!^2}. $$ There are other examples in the OEIS. In general, when you have a Hypergeometric function power series there can be $\,n!^2\,$ in the denominator of the terms. For example, $$ {}_2F_1(a,b;1;x) = \sum_{n=0}^\infty \frac{(a)_n(b)_n} {n!^2}x^n. $$