  In this paper, we find determinant formulas of several Hessenberg-Toeplitz
matrices whose nonzero entries are derived from the small and large Schroder
and Fine number sequences. Algebraic proofs of these results can be given which
make use of Trudi's formula and the generating function of the associated
sequence of determinants. We also provide direct arguments of our results that
utilize various counting techniques, among them sign-changing involutions, on
combinatorial structures related to classes of lattice paths enumerated by the
Schroder and Fine numbers. As a consequence of our results, we obtain some new
formulas for the Schroder and Catalan numbers as well as for some additional
sequences from the OEIS in terms of determinants of certain Hessenberg-Toeplitz
matrices.
