p^q-Catalan Numbers and Squarefree Binomial Coefficients

Abstract

In this paper we consider the generalized Catalan numbers F(s,n)= 1/((s-1)n+1) binom{sn}{n}. We find all $n$ such that for $p$ prime, p^q divides F(p^q,n), q>=1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers. We also prove that \binom{p^qn+1}{n}, p^q<=99999, is squarefree for n sufficiently large (explicit), and with the help of the generalized Catalan numbers we find the set of possible exceptions. As consequences, we obtain that binom{4n+1}{n}, binom{9n+1}{n} are squarefree for n> 2^{1518}, respectively n>3^{956}, with at most 2^{18.2}, respectively 3^{15.3} possible exceptions.