A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal

We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=\mathbb{k}[x_1, \dots, x_n]$, where $\mathbb{k}$ 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 $(x_1^2, \dots, x_n^2)$. 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 $G$.