Find algorithm or generator formula with automated procedure and pattern recognition

I am trying to find a generator formula for the discriminant of a polynomial and believe it would be an interesting problem for the Mathematica community to know how to automate evaluation of certain terms to discover an algorithm.

I am looking at the discriminant evaluated by

parnum = 4;
parint = 1;
(m = Array[Subscript[a, ##] &, {parnum, parnum}]);
(m = m /. Subscript[a, i_, j_] :> Subscript[a, j, i] /; j < i) //
MatrixForm;
b = Array[Subscript[be, # - 1] &, {parnum, 1}];
b = b /. {Subscript[be, 0] -> 1,
Subscript[be, k_] :> (-1)*Subscript[be, k] /; k > 0};
p = Transpose[b].m.b;
Collect[(1/4)*Discriminant[p, Subscript[be, parint]], Subscript[be, 2]]
(mi = Minors[m, 2]);
Minors[m, 2] // MatrixForm;


which gives me a certain discriminant \begin{eqnarray} -M_{11} - \beta_{2}^{2} M_{44} - 2 \beta_{2} M_{14} - \beta^{2}_{3} M_{55} - 2 \beta_{3} M_{15} - 2 \beta_2 \beta_3 M_{45}. \end{eqnarray} The $M_{i,j}$ expressions correspond to entries of the matrix of second minors.

Now, I am wondering if you guys could help me use FindSequenceFunction or a related function to somehow build code that allows me increase the dimensionality of the matrix denoted by parnum and the parameter I am solving for denoted by parint and also to recognize the ensuing pattern?

The above discriminant then follows a certain pattern that always involves the same generating formula, which can be written as a series expansion or rather four separate ones as outlined below.

Now similar approaches have been discussed here for a polynomial and here for integration but here I would like to have my discriminants expressed as a series of elements of the matrix of second minors.

One can quickly note that the structure of the formula is quite simple. It consists of

1. an intercept
2. a quadratic term in all the parameters I am not solving for
3. an interaction term, i.e. multiplication of all parameters
4. and linear terms for each parameter.
• Perhaps a good start would be to just generate a sequence for a five by five matrix and then split the elements into linear, quadratic, and intercept as well as interaction terms and find separate sequences for those? Commented Mar 1, 2015 at 19:30