Generalized Catalan Numbers: Linear Recursion and Divisibility
Indian Statistical Institute
8th Mile Mysore Road
We prove a linear
recursion for the generalized Catalan
As a consequence, we show
and only if
for all integers
This is a generalization of the well-known result that the usual
is odd if and only if
is a Mersenne
. 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
Full version: pdf,
(Concerned with sequence
Received May 21 2009.
revised version received October 28 2009.
Published in Journal of Integer Sequences, November 4 2009.
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