Combinatorial interpretation of composition of power series? 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 $a_n$?

 A: 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.
A: It seems solutions of the functional equation $g(g(t))=t$ are related to pseudo-involutions of the Riordan group. See http://www.sciencedirect.com/science/article/pii/S0166218X0900016X (Riordan group involutions and the $\Delta$-sequence, by Gi-Sang Cheon, Sung-Tae Jin, Hana Kim and Louis W. Shapiro). From the paper we can get about ten more solutions of this functional equation (Table 1 in the paper). Note that $g(t)=-f(t)$, where $f(t)$ is the generating function from Table 1. The simplest example is $f(t)=\frac{t}{1-t}$ related to the Pascal triangle.
A: When rephrased (using the positive generating series $g=-f$) as the identity $g(-g(-t))=t$, this suggests that there may exist a quadratic nonsymmetric operad with this generating series, which is Koszul and whose Koszul dual is itself.
I do not know such an operad. Looking at Loday's encyclopedia of operads, a possible candidate could have been the operad of Malcev algebras, but Loday only gives the first three terms $1,3,9$, which is probably not enough to draw any conclusion. And moreover, this is not a nonsymmetric operad. It may be possible to try to find one, maybe being guided by the existing combinatorics.
To illustrate, the simpler case of $g(t)=t/(1-t)$ is related to the Associative operad, which is Koszul and self-dual.
