The $d(n):=D(n,n)$ is OEIS sequence [A005773](http://oeis.org/A005773). The Hankel determinant property is given in the sequence entry. Also the recursion $\;nd_{n}=2nd_{n-1}+3(n-2)d_{n-2}$. A proof could come from a similar proof of the Hankel determinant property of the Catalan numbers. A related property is to use $d_0=1$ and take $\det(H_n^0)$ with $d_0$ in the first row and column giving OEIS sequence [A163806](http://oeis.org/A163806) $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the [Lindstrom-Gessel-Viennot Lemma](https://en.wikipedia.org/wiki/Lindstr%C3%B6m%E2%80%93Gessel%E2%80%93Viennot_lemma). This can be used to show that the unique solution to $\det(H_n^0)=\det(H_n^1)=1$ for all $n$ is the Catalan numbers. For a reference look at Aigner, Catalan-like numbers and determinants, J. Comb. Th. A 87 (1999), 33-51.

By the way, notice how the diagram for $D(4,4)=13$ is
$$
\begin{matrix}
   1 & 2 & 5 & 13 \\
   1 & 3 & 8 & 21 \\
   1 & 3 & 8 & 21 \\
   1 & 2 & 5 & 13  
\end{matrix}‎‎$$
where each entry is the sum of two or three entries in the preceding column.

Proofs of this and related examples are given by Johann Cigler in [Some nice Hankel determinants](http://homepage.univie.ac.at/johann.cigler/preprints/hankel-conjectures.pdf).