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In this paper, we find determinant formulas of several Hessenberg-Toeplitzmatrices whose nonzero entries are derived from the small and large Schroderand Fine number sequences. Algebraic proofs of these results can be given whichmake use of Trudi's formula and the generating function of the associatedsequence of determinants. We also provide direct arguments of our results thatutilize various counting techniques, among them sign-changing involutions, oncombinatorial structures related to classes of lattice paths enumerated by theSchroder and Fine numbers. As a consequence of our results, we obtain some newformulas for the Schroder and Catalan numbers as well as for some additionalsequences from the OEIS in terms of determinants of certain Hessenberg-Toeplitzmatrices.

Hessenberg-Toeplitz Matrix Determinants with Schroder and Fine Number Entries