Regarding a Paper by Paul.A Clement on Tridiagonal Matrices In Paul.A Clement's (1959) paper:
A Class of Triple-Diagonal Matrices for Test Purposes

SIAM Review, Vol. 1, No. 1 (Jan., 1959), pp. 50-52

He makes the claim that the eigenvalues of :
$ \begin{pmatrix}
0 & y_{1} & 0 & ... & 0 \\\ 
x_{1} & 0 & y_{2} &  & ... \\\
0 & x_{2} & 0 & ... & 0 \\\ 
... &  & ... &  & y_{n-1} \\\ 
0 & ... & 0 & x_{n-1} & 0
\end{pmatrix} $
are $\pm (n), \pm (n-2), ..., (\pm1 \; or \; 0)$ for $x_{k} = k$ and $y_{k} = n-k+1$.
Specifically, and I quote, "then a theorem of Sylvester establishes that the eigenvalues of this An+, are the
numbers".
I can't for the life of me figure out what theorem and/or how it follows from them. I am familiar with Sylvester's formula for matrices in terms of their eigenvalues, but to get Frobenius covariants of a matrix A one needs to know the eigenvalues to start with.
Am I overlooking something trivial here?
 A: I think the claim in this form is wrong. The eigenvalues are not integral. For example, with $n=4$ the matrix is
$$
A=\begin{pmatrix}
0 & 4 & 0 & 0\cr
1 & 0 & 3 & 0\cr
0 & 2 & 0 & 2 \cr
0 & 0 & 3 & 0
\end{pmatrix}.
$$
The characteristic polynomial of this matrix is 
$\chi (t)=t^4-16 t^2 +24$, which has no integral roots.
Am I overlooking something ?
Edit: just visited the site http://math.nist.gov/MatrixMarket/deli/Clement/
Here I saw that the upper diagonal must be $3,2,1$, not $4,3,2$. Then everything is OK.
So $y_k=n-k$ rather than $y_k=n-k+1$, what you wrote. We have a recursion for $\chi(t)=\det(t\cdot id-A)$, see Eigenvalues of Symmetric Tridiagonal Matrices.
Edit: A proof for the correct claim can be found in the paper of Taussky and Todd, "Another look at a matrix of Mark Kac", Linear Algebra Appl. 150 (1991), 341-360.
More details are also at
https://math.stackexchange.com/questions/405670/regarding-a-paper-by-paul-a-clement-on-tridiagonal-matrices, which was remarked by Gerry (thank you ). Also, Darij's remark is correct, that the proof does not follow easily, so I have edited this here.
