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The $d(n):=D(n,n)$ is OEIS sequence 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 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. 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.

The $d(n):=D(n,n)$ is OEIS sequence 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 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. 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.

Catalan and related sequence proofs are given by J, Cigler in Some nice Hankel determinants.

The $d(n):=D(n,n)$ is OEIS sequence 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 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. 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.

The $d(n):=D(n,n)$ is OEIS sequence 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 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. 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.

The $d(n):=D(n,n)$ is OEIS sequence 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 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. 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 2 or three entries in the preceeding column.

The $d(n):=D(n,n)$ is OEIS sequence 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 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. 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.