It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix

$ \begin{pmatrix} 0 & n-1 & 0 & ... & 0 \\\ 1 & 0 & n-2 & & ... \\\ 0 & 2 & 0 & ... & 0 \\\ ... & & ... & & 1 \\\ 0 & ... & 0 & n-1 & 0 \end{pmatrix}$

are $\pm (n), \pm (n-2), ..., (\pm1 \; or \; 0)$. Please refer to Regarding a Paper by Paul.A Clement on Tridiagonal Matrices and http://math.stackexchange.com/questions/405670/regarding-a-paper-by-paul-a-clement-on-tridiagonal-matrices for details.

My question is do we know the eigenvalues of the leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix $(r>2)$:

$ \begin{pmatrix} 0 & n-1 & 0 & ... & 0 \\\ 1 & 0 & n-2 & & ... \\\ 0 & 2 & 0 & ... & 0 \\\ ... & & ... & & n-r+1 \\\ 0 & ... & 0 & r-1 & 0 \end{pmatrix}$

or do we have any lower/upper bound on the eigenvalues of this submatrix.