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:


1. Has this been observed before?
2. Are there any combinatorial interpretations of this identity in terms of the combinatorial descriptions of the sequence $a\_n$?



Combinatorial interpretation of composition of power series?
Basically $f(f(t)) = t$ is related to the fact that $y = f(t)$ and $t$ satisfy an algebraic relation $$y^2 t^2 - 3 y t + y + t = 0$$ which is symmetric in $y$ and $t$ and its graph passes through $(0,0)$ tangent to the line $y=-t$. That relation corresponds to the identity
$$ \sum\_{j=1}^{n-1} a\_j a\_{n-j} + 3 a\_{n+1} - a\_{n+2} = 0$$
I don't know if this has a combinatorial interpretation.


