A Generalization of Planar Pascal’s Triangle to Polynomial Expansion and Connection with Sierpinski Patterns

The very well-known stacked sets of numbers referred
to as Pascal’s triangle present the coefficients of the binomial
expansion of the form (x+y)n. This paper presents an approach (the
Staircase Horizontal Vertical, SHV-method) to the generalization of
planar Pascal’s triangle for polynomial expansion of the form
(x+y+z+w+r+⋯)n. The presented generalization of Pascal’s triangle
is different from other generalizations of Pascal’s triangles given in
the literature. The coefficients of the generalized Pascal’s triangles,
presented in this work, are generated by inspection, using embedded
Pascal’s triangles. The coefficients of I-variables expansion are
generated by horizontally laying out the Pascal’s elements of (I-1)
variables expansion, in a staircase manner, and multiplying them with
the relevant columns of vertically laid out classical Pascal’s elements,
hence avoiding factorial calculations for generating the coefficients
of the polynomial expansion. Furthermore, the classical Pascal’s
triangle has some pattern built into it regarding its odd and even
numbers. Such pattern is known as the Sierpinski’s triangle. In this
study, a presentation of Sierpinski-like patterns of the generalized
Pascal’s triangles is given. Applications related to those coefficients
of the binomial expansion (Pascal’s triangle), or polynomial
expansion (generalized Pascal’s triangles) can be in areas of
combinatorics, and probabilities.

Authors:

• Wajdi Mohamed Ratemi

Keywords:

• Generalized Pascal’s triangle
• Pascal’s triangle
• polynomial expansion
• Sierpinski’s triangle
• staircase horizontal vertical method.

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