This problem is [essentially the same as this one][1]. 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.


  [1]: http://mathoverflow.net/questions/156090/eigenvectors-of-a-particular-transition-matrix/156202#156202