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It's unlikely that these will have any combinatorial applications since the product of two "Bessel generating functions" with integer coefficients won't in general have integer coefficients (by which I mean the coefficients of $x^n/n!(n+k)!$).
I would say that a lot is known about the count of unlabeled trees; it's just more complicated than for labeled trees, and this is very typical of graphical enumeration. You can find lots of examples in Harary and Palmer's book "Graphical Enumeration." There is even an explicit though complicated formula (as a sum over partitions) for the number of unlabeled trees. It might be noted that there are some problems, such as counting self-complementary graphs, in which the unlabeled version has been solved, but not the labeled version.
The Robinson-Schensted correspondence restricted to FPFI is a bijection onto standard Young tableaux with every column of even length that preserves descents (i is a descent of a standard Young tableau if i+1 is to the left of i).
I believe that this identity is a limit of a finite sum that can be proved using the WZ method. Whether that makes it combinatorial is a matter of opinion.
The identity is just a special case of Vandermonde's theorem: \begin{align*}\sum_{i=1}^{j+1} (-1)^{i-1}\binom{n+i-1}{i-1}\binom{n+j+1}{j+1-i} & = \sum_{i=1}^{j+1} \binom{-n-1}{i-1}\binom{n+j+1}{j+1-i}\\ & = \binom{j}{j}=1. \end{align*}
There is a very nice combinatorial interpretation of identities $g(-f(-t))=t$ in terms of trees, which is related to operas (which I don't know anything about), due by S. Parker (but not published) and rediscovered by J.-L. Loday. (See also R. Bacher and R. Bacher and G. Schaeffer.) However, I couldn't get it to work for this problem.
This was observed by Michael Somos in 2004, as noted in the OEIS entry. As also noted in the OEIS entry, $f(t) = - tc(t)^3$, where $c(t)$ is the generating function for Catalan numbers. More generally, the compositional inverse of $xc_r(x^a)^b$ is $xc_{ab-r+1}(-x^a)^b$, where $c_r(x)$ is the generalized Catalan number generating function satisfying $c_r(x) = 1+xc_r(x)^r$; the OP's formula is the case $a=1$, $b=3$, $r=2$.
