The Column Addition Triangle (CAT) and Polynomials (Part II)

Polynomial Division

Polynomials of the type (x + y)ⁿ may be expanded using the row numbers in the Pascal triangle as the coefficients of the expanded equation. For example, using y=1 and the third row of the Pascal triangle (1 3 3 1) produces the equation x³ + 3x² + 3x + 1. The CAT triangle, however, discussed in Part I and shown below in the irregular triangle array format, can be constructed using an (x + 1) expression.

The numbers in each row of Table I correspond to the coefficients of a polynomial whose leading degree exponent is odd, so that, for example, rows 3 and 5 correspond to the equations:

An interesting property for all the equations generated by the CAT triangle is that each is divisible by x + 1 producing the irregular triangle with the Sloane number A004737 which means that the CAT triangle is of the type (x + 1) × (A polynomial derived from a row in A004737):

CAT and A004737

where the numbers in each row correspond to the coefficient of a polynomial whose leading degree exponent is even, and where, for example, rows 3, 4, 5 and 6 correspond to the equations:

The code for generating A004737 is similar to the previous code for generating the CAT triangle. See file text A004737-text. The output to this program, A004737-code when n=19, shows that the rows consists of both even and odd coefficients with an odd number of coefficients per row, unlike the CAT triangle (compare Tables I and II).

Those equations with odd number middle coefficients, 3 and 5, respectively, upon further division by x + 1 leave a remainder of 1, while those with even middle coefficients, 4 and 6, leave no remainder. Table III displays the results of dividing each polynomial derived from Table II by x + 1 where R1 stands for remainder of 1:

Three additional Triangles Exhibiting Ascending Diagonal Properties Part III are to be found in the next section. These triangles may be generated from the original A004737 (just as the CAT triangle may be generated) via multiplication of each row of A004737 by x+1 similar to the method used in the generation of rows in the Pascal triangle.

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