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Hankel matrices are square matrices whose coefficients $a_{i,j}$ only depend on the sum $i+j$. They appear in particular in the theory of orthogonal polynomials on the real line.
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Hankel determinants of binomial coefficients
For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & h_{2} & \dots & …