I am trying to calculate an upper bound for the determinant of a matrix of the form
$\begin{pmatrix}a_{n-1} & a_{n-2} & a_{n-3} & \cdots & a_{n-m+1} & a_{n-k+1}\\ a_{n} & a_{n-1} & a_{n-2} & \cdots & a_{n-m+2} & a_{n-k+2}\\ 0 & a_{n} & a_{n-1} & \cdots & a_{n-m+3} & a_{n-k+3}\\ \vdots & & \ddots & \ddots & & \vdots \\ 0 & 0 & \cdots & \cdots & a_n & a_{n-k+m}\end{pmatrix}$
which is an $m\times m$ upper-Hessenberg matrix, and a Toeplitz matrix but for a single column.
By Cramer's rule, the determinant of this matrix is the determinant of a Toeplitz-Hessenberg matrix (for which Trudi's formula gives a value) times one component of a solution of a linear system of equations. But I'm just not putting the pieces together. Please advise.