p-Catalan Numbers and Squarefree Binomial Coefficients

In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p divides F (p, n), and also determine all distinct residues of F (p, n) (mod p), q ≥ 1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if p ≤ 99999, then ( pn+1 n ) is not squarefree for n ≥ τ1(p) sufficiently large (τ1(p) computable). Moreover, using the results of the first part, we find n < τ1(p) (in base p), for which ( pn+1 n ) may be squarefree. As consequences, we obtain that ( 4n+1 n ) is squarefree only for n = 1, 3, 45, and ( 9n+1 n ) is squarefree only for n = 1, 4, 10.