Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

KALLOS, G., The generalization of Pascal's triangle from algebraic point of view

The generalization of Pascal's triangle from algebraic point of view GÁBOR KALLÓS* Abstract. In this paper we generalize Pascal's Triangle and examine the connections between the generalized triangles and powering integers and polynomials respectively. The interesting and really romantic Pascal's Triangle is a favourite research field of mathematicians for a very long time. The table of binomial coefficients has been named after Blaise Pascal, a FYench scientist, but was known already by the ancient Chinese and others before Pascal (Edwards [i]). Among the elements of the triangle a lot of interesting connections exist. One of them is that from the n-th row of the triangle with positional addition we get the n-th power of 11 (Figure 1.), where n is a nonnegativ integer, and the indices in the rows and columns run from 0. 1 1 1 12 1 13 3 1 1 4 6 4 1 1 5 10 10 5 1 1 = 11°, 11 = ll

1, 121 = ll

2, 1331 = ll

3, 14641 = ll

4, 161051 = ll

5,... Figure 1: The powers of 11 in Pascal's triangle This comes immediately from the binomial equality (o)

i

o"

+G)

+Q

i

o"2

+• • •

+(„! i)

101

+(I) i

o°

= An interesting way of generalizing is if we construct triangles in which the powers of other numbers appear. To achieve this, let us consider Pascal's Triangle as the 11-based triangle, and take the following. This paper was completed during the stay of the author at Paderborn University, Germany in summer 1996.