Journal of Integer Sequences, Vol. 12 (2009), Article 09.7.5 |

Stat-Math Unit

Indian Statistical Institute

8th Mile Mysore Road

Bangalore 560059

India

**Abstract:**

We prove a *linear* recursion for the generalized Catalan
numbers
when .
As a consequence, we show
if
and only if
for all integers .
This is a generalization of the well-known result that the usual
Catalan number is odd if and only if is a Mersenne
number . Using certain beautiful results of Kummer and
Legendre, we give a second proof of the divisibility result for
. We also give suitably formulated inductive proofs of
Kummer's and Legendre's formulae which are different from the
standard proofs.

(Concerned with sequence A000108.)

Received May 21 2009.
revised version received October 28 2009.
Published in *Journal of Integer Sequences*, November 4 2009.

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