\documentclass[12pt]{article}
\begin{document}
\begin{abstract}
We prove a {\it linear} recursion for the generalized Catalan
numbers $C_a(n) := \frac{1}{(a-1)n+1} {an \choose n}$ when $a \geq
2$.
As a consequence, we show $p \, | \, C_p(n)$ if
and only if $n \neq \frac{p^k-1}{p-1}$ for all integers $k \geq 0$.
This is a generalization of the well-known result that the usual
Catalan number $C_2(n)$ is odd if and only if $n$ is a Mersenne
number $2^k-1$. Using certain beautiful results of Kummer and
Legendre, we give a second proof of the divisibility result for
$C_p(n)$. We also give suitably formulated inductive proofs of
Kummer's and Legendre's formulae which are different from the
standard proofs.
\end{abstract}
\end{document}