While I was toying around with matrices, I chanced upon a family of tridiagonal matrices $M_n$ that take the following form: the superdiagonal entries are all $1$'s, the diagonal entries take the form $m_{j,j}=4 j (2 n + 3) - 8 j^2 - 6 n - 5$, and the subdiagonal entries take the form $m_{j+1,j}=4 j (2 j - 1) (2 n - 2 j + 1) (n - j)$.
For example, the $4\times 4$ member of this family looks like this:
$$M_4=\begin{pmatrix} 7 & 1 & 0 & 0 \\ 84 & 27 & 1 & 0 \\ 0 & 240 & 31 & 1 \\ 0 & 0 & 180 & 19\end{pmatrix}$$
I checked the eigenvalues of members of this family and I found that each member has the squares of the first few odd integers as eigenvalues. (For example, the eigenvalues of $M_4$ are $1,9,25,49$.) I couldn't find a way to prove this though.
I wish someone would help me! Thanks!