The Catalan numbers have a famous generalization associated to finite irreducible reflection groups. Afaik, the formula $$\operatorname{Catalan}(W)=\prod_{i=1}^n \frac{d_i+h}{d_i}$$ appeared first in the paper "Noncrossing partitions for classical reflection groups" (Discrete Math., 1996) by Vic Reiner. Here, $d_1,\dots,d_n$ are the invariant degrees and $h$ is the Coxeter number.
I believe the first appearance of these numbers (though not yet with a uniform formula) in general simply-laced types was in the paper "Quotients of representation-finite algebras" (Communications Alg., 1987) by Gabriel and J.A. de la Peña as the number of "discrete subsets" of the path algebra of the Dynkin quiver. These discrete subsets are now known to be counted by the Catalan numbers (reference missing for now). They counted the discrete subsets as the Catalan numbers
- of type $A_n$ correctly on page 292 as $\frac{1}{n+2}\binom{2n+2}{n+1}$,
- of type $D_{n+1}$ correctly on page 293 as some longer formula that could be simplyfied to $\binom{2n+2}{n+1}-\binom{2n}{n}$,
- of types $E_6$ and $E_7$ not correctly on page 294 as $468$ in $E_6$ and $4159$ in $E_7$. The correct counts would have been $833$ in type $E_6$ and $4160$ in type $E_7$.
- of type $E_8$ correctly on page 294 as $25080$.