Analytic aspects of Delannoy numbers

The Delannoy numbers $d(n,k)$ count the number of lattice paths from $(0,0)$ to $(n?k,k)$ using steps $(1,0),(0,1)$ and $(1,1)$. We show that the zeros of all Delannoy polynomials $d_n\left(x\right)={\sum }_{k=0}^{n}d(n,k)x^k$ are in the open interval $(?3?2\sqrt{2},?3+2\sqrt{2})$ and are dense in the corresponding closed interval. We also show that the Delannoy numbers $d(n,k)$ are asymptotically normal (by central and local limit theorems).