Last time, I demonstrated my own definition of an integer polynomial pseudo-derivative and how it can be used to construct the results of the polynomial using only sums. Now I wish to look closer at some of the columns in the pseudo-derivative tree of the same polynomial: p(x) = ax^3 + bx^2 + cx + d.
Looking back at the previous post, there are four columns in the result portion of the pseudo-derivative tree (the supplied “x” variable is not counted as a result column).
Column 4 is the final nonzero pseudo-derivative for this polynomial, and the column has only 6a in every entry. Column 3 is more interesting, as it begins with 6a+2b and each successive entry is increased by 6a. Column 2 is the one I wish to focus on now, and in particular the “a” term of each entry. After reducing to just the “a” term, the column looks like this:
The differences between these terms have already been demonstrated, but now let’s look at the similarities. In particular, subtract a from all terms, then divide all terms by 6a:
The term after 37a is 61a, which reduces to 10 in the final listing. This further reinforces what is probably obvious: this sequence is “diagonal row 2” of Pascal’s Triangle, with generating formula k(k+1)/2, which is also the famous set of Triangle numbers. In fact, column 3 corresponds to “diagonal row 1” of Pascal’s Triangle, and if p(x) were degree 4, we would have a column which corresponds to “diagonal row 3” of Pascal’s Triangle.
This correspondence continues through the entire range of Pascal’s Triangle given polynomials of increasing degree.