For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote by $H_{n}$ the $n\times n$ Hankel matrix of the form $$ H_{n}:=\begin{pmatrix} h_{0} & h_{1} & \dots & h_{n-1}\\ h_{1} & h_{2} & \dots & h_{n}\\ \vdots & \vdots & \ddots & \vdots\\ h_{n-1} & h_{n} & \dots & h_{2n-2} \end{pmatrix}. $$ Just by a mathematical curiosity, I am interested in formulas for $\det H_{n}$ where $$ h_{n}=\binom{pn}{n} $$ where $p\geq2$ is an integer.

If $p=2$, the Hankel determinant (also called the Hankel transform) of central binomial coefficients is well-known and is not very difficult to evaluate. It reads $$ \det H_{n}=2^{n-1}. $$

For $p=3$, the formula (and its derivation) is more complicated, though it is also known that $$ \det H_{n}=3^{n-1}\left(\prod_{i=0}^{n-1}\frac{(3i+1)(6i)!(2i)!}{(4i)!(4i+1)!}\right), $$ see, for example, this paper for the proof.

Nevertheless, if $p\geq4$, I have not found any result about the corresponding Hankel determinant, nor I was able to even guess a form of the possible identity for $p=4$ while experimenting with Mathematica.

**My questions are as follows:**

- Is there something known about $\det H_{n}$ for $p\geq4$?
- Is anybody able to at least guess a formula for the case $p=4$?
- Is there any reason which might indicate that an explicit formula for $\det H_{n}$, with $p\geq4$, (like the one for $p=3$) need not exist?

**Remark:** I would like to remark that I know Krattenthaler's survey on advanced determinant calculus but I didn't find an answer therein.

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