This is a minor curiosity that came up in a joint project recently.

Consider the sequence $a_n=3\frac {(2n)!}{(n+2)!(n-1)!}$ (A000245 in OEIS). It has multiple combinatorial descriptions.

One can write the corresponding generating function, but I will do it with an extra sign. $$ f(t) = -\sum_{n=0}^{\infty} a_n t^n = -t-3t^2-9t^3-28t^4-90t^5 -... $$ which has a simple though inelegant description $$ f(t) = \frac {(1-t)\sqrt{1-4t}-1+3t}{2t^2}. $$

So far it is all perfectly boring. But the interesting thing is that this $f$ satisfies the identity $$ f(f(t))=t. $$ My questions are:

- Has this been observed before?
- Are there any combinatorial interpretations of this identity in terms of the combinatorial descriptions of the sequence $a_n$?

Enumerative Combinatorics, vol. 1, Exercise 1.168. – Richard Stanley Jul 17 '14 at 1:13