pq-Catalan numbers and squarefree binomial coefficients Original Research Article

In this paper, we consider the generalized Catalan numbers image, which we call s-Catalan numbers. For p prime, we find all positive integers n such that pq divides F(pq,n), and also determine all distinct residues of F(pq,n)(mod pq), qgreater-or-equal, slanted1. 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 pqless-than-or-equals, slant99 999, then image is not squarefree for ngreater-or-equal, slantedτ1(pq) sufficiently large (τ1(pq) computable). Moreover, using the results of the first part, we find n<τ1(pq) (in base p), for which image may be squarefree. As consequences, we obtain that image is squarefree only for n=1,3,45, and image is squarefree only for n=1,4,10.