Novus Pascalloidia:

If the following is true: A=1; B=1; C=1; D=1; E=1, etc., then the coefficients ' values matter only, and the terms drop away. Pascalloids as pure number tiling patterns holds true in this case.

If instead A=A; B=B; C=C; D=D; E=E, etc., where there are combinatoric solutions, there will be axis and indices, where there are axis and indices there will be constructability of some terms to go with the polynomials

Not to be omitted from this study is a version of Pascal's Square that expands into a 4-sided pyramid (if you count each sheet of numbers as a row), as a 5th side is just a row. This is as easy to discover as it is just one dimension more complicated than Pascal's Squarele. We will operate on it in 2-dimensional sheets. Depicted is the algorithm that is repeated in order to generate each successive square. This algorithm is repeated "super-positionally", digitally, over a complete tiling pattern. Here, the ambient grid of zeros is represented by empty space outside the square of numbers. This one (pattern / Pascalloid) I had stumbled upon easily in 1998 while taking Physics at Evergreen State College..

Note that all terms have collapsed to unity.
This means that each of the four variables A,B,C & D is set to 1.

algebraic:  ((A+B)(C+D))N


     (N!)         (N!)    
X!(N-X)! Y!(N-Y)!
Download XL file:
Square Computation Simplified Alg.xls

Square Number Crystal

N = 0
N = 1
N = 2
N = 3
N = 4
N = 5
N = 6
N = 7
N = 8
N = 9
N = 10
N = 11
N = 12
N = 13
N = 14
N = 15
N = 16
N = 17
N = 18
N = 19
N = 20
N = 21
N = 22
N = 23
N = 24
N = 25
N = 26
N = 27