Two integer sequences related to Catalan numbers

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 + ∑ j = 1 n − 1 ( − 1 ) j ( 2 n − 1 2 j − 1 ) A j C n − j and a n = 2 A n / C n . Then a n (hence A n ) is a positive integer.