Generalized Catalan Numbers: Linear Recursion and Divisibility

We prove a linear recursion for the generalized Catalan numbers Ca(n) := 1 (a−1)n+1 ( an n ) when a ≥ 2. As a consequence, we show p |Cp(n) if and only if n 6= p−1 p−1 for all integers k ≥ 0. This is a generalization of the well-known result that the usual Catalan number C2(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 Cp(n). We also give suitably formulated inductive proofs of Kummer’s and Legendre’s formulae which are different from the standard proofs.