Explicit inversion formulas for properly Hessenberg and Toeplitz-Hessenberg matrices

The nice inversion properties of Toeplitz-Hessenberg matrices from a Hessenbergian representation of Catalan's numbers encourage us to provide explicit inversion formulas (in terms of basic arithmetical operations involving entries from the original matrix) for the non-singular Toeplitz-Hessenberg matrices. Our approach is based on an elementary matrix inflation method, joint with the nested sums formulas. These explicit inversion formulas are then extended to apply to the entries of the inverse of every non-singular properly Hessenberg matrix.