Positive definite matrices and Catalan numbers

Abstract:

It is shown that the number of $n \times n$ integral triple diagonal matrices which are unimodular, positive definite and whose sub and super diagonal elements are all one, is the Catalan number $(_n^{2n})/(n + 1)$. More generally, it is shown that if A is a fixed integral symmetric matrix and d is a fixed positive integer, then there are only finitely many integral diagonal matrices D such that $A + D$ is positive definite and $\det (A + D) = d$.