2000/10/15 by Pantelimon Stănică, Pantelimon Stanica, Stanica, Pantelimon
Mathematics · #11B65 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B65
paper · pdf · doi:10.48550/arxiv.math/0010148
18 pages
arxiv created 2000/10/15 · openalex publication_date 2000/10/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \binompqn+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.