pq-Catalan Numbers and Squarefree Binomial Coefficients

الأصل · EN

In this paper we consider the generalized Catalan numbers F(s,n)= 1/((s-1)n+1) binomsnn. We find all n such that for p prime, pq divides F(pq,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 pqn+1n, pq<=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 binom4n+1n, binom9n+1n are squarefree for n> 2¹⁵¹⁸, respectively n>3⁹⁵⁶, with at most 2¹⁸.², respectively 3¹⁵.³ possible exceptions.