Hankel determinant evaluation of special lattice paths ‎Let ‎$‎n‎$ ‎be a‎ ‎positive ‎integers and ‎$‎T=T_{n,n}‎$ ‎be the ‎$‎n\times n‎$‎ table in the first quadrant composed of $n^2$ unit squares‎, ‎whose $(x,y)$-blank is locate in the $x^{th}$-column from the left and the $y^{th}$-row from the bottom hand side of $T_{n,n}$ . 
‎
Put ‎$‎D(n,n)‎$ ‎be ‎the ‎number ‎of ‎all ‎lattice ‎path ‎from the ‎first ‎column to entry ‎$‎(n,n)‎$‎ ‎of the ‎table ‎‎$‎T‎$ which  steps comes from the set  $S=\{(1,0)‎, ‎(1,1),(1,-1)\}$.(we allowed to move only to the right (up, down or straight) ). ‎It ‎is ‎easy ‎to ‎see ‎for ‎‎$‎n\geq 2‎$‎
$$D(n,n)=D(n-1,n)+D(n-1,n-1)$$
‎where ‎$‎D(1,1)=1, D(2,2)=2, D(3,3)=5,‎ D(4,4)=13, \cdots‎‎$‎.
Notice, the entry $(x,y)$ means cordinate $x$ and $y$ in the table $T$ not the row $x$ and column $y$.
For example
$$D(3,3)=D(2,3)+D(2,2)=2+3=5$$ and
$$D(2,3)=D(1,3)+D(1,2)=1+1=2.$$
For calculating $D(2,3)$ you must consider the table with 3 rows and columns and by using this table calculate all lattice paths reach to entry $(2,3)$ in this table!!
in my arXiv paper, there are some references for this sequence!!
 I ‎think ‎these ‎lattice ‎paths ‎very ‎interesting ‎and ‎obtained ‎some ‎results ‎about ‎them. Put ‎$‎D(n,n)=d_n‎‎$, I check and known that the  Hankel determinant evaluation of ‎$‎D(n,n)‎$ is
$$
\det(H_n^1)=\det‎‎
\begin{bmatrix}
    d_{1} & d_{2} & d_{3} & \dots  & d_{n} \\
    d_{2} & d_{3} & d_{4} & \dots  & d_{n+1} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    d_{n} & d_{n+1} & d_{n+2} & \dots  & d_{2n-1}
\end{bmatrix}‎‎
=1.
$$‎‎‎‎
Do you have ideas or comments for proving it?
Thank you so much for any help or comment.
 A: 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.
