We examine the ideal I= (x₁², , xₙ², (x₁++xₙ) ²) in the polynomial ring Q=kx₁, , xₙ, where k is a field of characteristic zero or greater than n. We also study the Gorenstein ideal G linked to I via the complete intersection ideal (x₁², , xₙ²). We compute the Betti numbers of I and G over Q when n is odd and extend known computations when n is even. A consequence is that the socle of Q/I is generated in a single degree (thus Q/I is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal G.
Diethorn et al. (Fri,) studied this question.