TITLE: A generalization of Catalan numbers
QUESTION [7 upvotes]: It is well-known that the $n$th Catalan number is equal to $(n+1)^{-1}\binom{2n}{n}$. A long time ago I had wondered what happens if you look at the sequence generated by $(n+k)^{-1}\binom{pn}{n}$ - for which $p,k$ is it integral?
I found no other integral-producing values except for $(p,k)=(2,1)$ but then I'm probably missing something. (Also, MATLAB, which is my main tool, quickly runs out of precision on this kind of computation).
So - is this known? Trivial, perhaps?

REPLY [7 votes]: You might also be interested in the rational Catalan numbers, defined for $(a,b)=1$ as $\mathrm{Cat}(a,b) := \frac{1}{a+b}\binom{a+b}{a}$.
See these slides by Drew Armstrong for their significance: http://www.math.miami.edu/~armstrong/RCCinDC.pdf
As this paper explains, they are a further generalization of Fuss-Catalan numbers: http://arxiv.org/abs/1305.7286

REPLY [2 votes]: One more related oject is the Catalan's triangle with integer entries 
$$B(n,r)=\frac rn{2n\choose n-r}$$
(see also the free book Catalan Numbers With Applications - T Koshy (Oxford, 2009)