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Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form
$H_n(a_0(n))=\begin{pmatrix}
a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\
(1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\
(2!)^2 &...
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