Jacobsthal numbers and alternating sign matrices

Let A(n) denote the number of n n alternating sign matrices and J _m the m ^th Jacobsthal number. It is known that A(n) = n Π=0 (3l+1)!/(n+l)! The values of A(n) are in general highly composite. The goal of this paper is to prove that A(n) is odd if and only if n is a Jacobsthal number, thus showing that A(n) is odd in nitely often.