Let $x\_1, \ldots, x\_n$ be variables, $e\_n$ be the elementary symmetric polynomials. I will denote the discriminant by
$$D\_n(x\_1, \ldots, x\_n) = \prod\_{i<j} (x\_i - x\_j)^2$$
And a generalized discriminant by 
$$D\_k(x\_1, \ldots, x\_n) = \sum\_{S \subset \{1, \ldots, n\}, |S| = k} ~~~\prod\_{\{i,j\} \subset S} (x\_i - x\_j)^2$$


If $p$ is the monic polynomial with $x\_i$ as roots, then it is well known that
$$D\_n(x\_1, \ldots, x\_n) = R(p,p') = \left| \begin{matrix} n & (n-1) e\_1 & (n-2) e\_2 & \ldots & 0& 0 & 0 \\ 1 & e\_1 & e\_2& \ldots & 0 & 0 &0\\ 0 & n & (n-1)e\_1 & \ldots & 0 & 0 & 0\\ 0 & 1 & e\_1 & \ldots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & \ldots & 0& 1 & e\_1 & \ldots & e\_n \\ 0 & \ldots & 0 & 0 & n & \ldots & e\_{n-1} \end{matrix} \right|$$
(This matrix is formed by the row $(n , (n-1) e\_1, \ldots, e\_{n-1})$ and $(1, e\_1, e\_2, \ldots, e\_n)$ alternating and moving over to the right once each time). \ \


After testing multiple examples, I have hypothesized that $D\_k$ is the leading principal minor of this matrix of size $(2k+1) \times (2k+1)$. I'm sure this result must be known, so I was hoping for a proof or a reference.



Principal Minors of the Resultant
I've seen your $D\_k$ be called a *subdiscriminant*. Similarly the principal minors of the Sylvester matrix of two polynomials are the *subresultants*. The result you want is that subdiscriminants are (up to a constant, depending on the definition) equal to the corresponding subresultant of the polynomial and its derivative. A possible reference is the beginning of chapter 4 of ["Algorithms in Real Algebraic Geometry" by S. Basu, R. Pollack, M-F. Roy](https://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted3.html). Your result is Proposition 4.28.


