-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. We find all natural numbers n such that for p prime, p divides F (p, n), q ≥ 1 and all distinct residues of F (p, n) (mod p), q = 1, 2. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. We also prove that ( p q n+1 n ) , p ≤ 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 ( 4n+1 n ) , ( 9n+1 n ) are squarefree for n ≥ 2, respectively n ≥ 3, with at most 2, respectively 3 possible exceptions.