I'm concerned here with Catalan numbers.
There are many combinatorial interpretation of these numbers. Here I would focus on the interpretation around words build with 2 symbols, let say [ and ]. The Nth Catalan number for such pair of symbols gives the number of words of length 2N such that any sub words never contains more [ than ].
I'm interested what we can say if we take not only 2 symbols but 4 for example. Let say [, ] and (, ).
Then Nth "Generalized" Catalan number would be interpreted as the number of words of length 2N build with these 4 symbols such that any sub word is, let say "simple Catalan" for the pair [,], or "simple Catalan" for the pair (,) or both. In other words for any sub words, there is a pair of symbols which are "simple Catalan" for this sub word.
Hope I'm clear enough.
For two different types of parenthesis this the sequence is listed in the OEIS here.
Words with balanced $k$-type parentheses are known as $\text{Dyck}(k)$ words. Maybe this helps for further investigations.