This problem is essentially the same as this one. In particular, let $J$ be the anti-diagonal identity matrix, and $P^{-1}$ be the matrix mentioned in the link above. Then, the matrix in the current post is nothing but \begin{equation*} B_n = JP^{-T}J^T. \end{equation*} Since $J^TJ=I$, we can recover eigenvectors and values of $B_n$ using the derivation for $P$ in the linked post.

This problem is essentially the same as this one. In particular, let $J$ be the anti-diagonal identity matrix, and $P^{-1}$ be the matrix mentioned in the link above. Then, the matrix in the current post is nothing but \begin{equation*} B_n = JP^{-T}J^T. \end{equation*} Since $J^TJ=I$, we can recover eigenvectors and values of $B_n$ using the derivation for $P$ in the linked post.