For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not.

Recently, joint with my graduate student Han Wang, we proved in [arXiv:2206.12317][1] the identity 
$$D_1(n)=1+\frac{n^2(n^2-1)}{12}\tag{1}$$
via eigenvalues. (See also [A079034][0] at OEIS.)
Based on my numerical computations, I conjectured that
$$\begin{align}D_2(n)&=1+\frac{n^2(n^2-1)}{1080}(n^5-5n^3-36n^2+4n+54)
\\&=\frac{(n^2-4)(n^2+2n+3)(n^5-2n^4-n^3-28n^2+60n-90)}{1080}\end{align}\tag{2}$$
and
$$\begin{align}D_3(n)=&1+\frac{n^2(n^2-1)}{672000}
\\&\times(n^{12}-19n^{10}+123n^8-337n^6+12376n^4-44144n^2+40000)\end{align}\tag{3}$$
(cf. [A355175][2] and [A355326][3] at OEIS). Note that the equation 
$$x^5-2x^4-x^3-28x^2+60x-90=0$$
over $\mathbb Q$ is not solvable by radicals.

In view of $(1)-(3)$, I have the following general conjecture.

**Conjecture.** Let $m$ be any positive integer. Then $D_m(n)$ has the form
$1+n^2(n^2-1)P_m(n)$, where $P_m(n)$ is a polynomial in $n$ with rational number coefficients whose degree is $(m+1)^2-4$.

**QUESTION.** How to prove the formulas $(2)$ and $(3)$? Is the above general conjecture true?

Your comments are welcome!


[1]: http://arxiv.org/abs/2206.12317 
[0]: https://oeis.org/A079034
[2]: https://oeis.org/A355175
[3]: https://oeis.org/A355326