I came a cross with the following number $$ C_{n}(m) = \frac{1}{n+m+1}\binom{m+2n}{n} $$ where $m, n \in \mathbb{N}$. When $m=0$, it is a well-known Catalan number.
I am curious about whether this number is somewhat known/studied in somewhere? If yes, I would like to take a look on it.
I am trying to search this quantity but what I've got so far so-called 'super-Catalan' number which is not related...
Any comments/suggestions will be very appreciated.
[Edited: $\frac{1}{n+1} \to \frac{1}{n+m+1}$.]