Two integer sequences related to Catalan numbers

Abstract

We prove the following conjecture of Zeilberger. Denoting by $C_n$ the Catalan number, define inductively $A_n$ by $(-1{)}^{n-1}{A}_n=C_n+{\sum }_{j=1}^{n-1}(-1{)}^j\left(\genfrac{}{}{0ex}{}{2n-1}{2j-1}\right)A_j{C}_{n-j}$ and $a_n=2A_n/C_n$. Then $a_n$ (hence $A_n$) is a positive integer.

15 pages, LaTeX, to appear in Journal of Combinatorial Theory, Series A