Determinant formulas of some Toeplitz–Hessenberg matrices with Catalan entries

Abstract

In this paper, we consider determinants of some families of Toeplitz-Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.