A few years ago, I wanted to cite a result in a paper, for which I could not find a reference. I ended up not using the full strength of it, and the part that I needed could be easily proved. Still, I'd like to know where the full version appears. The result is as follows:

The eigenvalues of the matrix $$\left[\binom{i+j}{i}\binom{2n-i-j}{n-i}\right]_{0\le i,j\le n}$$ are $$\binom{2n+1}{k}, \qquad 0\le k\le n.$$

If a formula for the corresponding eigenvectors also appears somewhere, that would be helpful, too. (I only needed the fact that $\binom{2n+1}{n}$ has eigenvector $[1,1,\dots,1]^T$, and that's easy to see.) Thanks.