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 15 by heart. It is similar to some methods of by heart calculate artists (Surányi [5]). In Proposition 3 we have seen a connection of the aó-based triangle with the binomial theorem. Thus, we expect for the 00^1^2 • • -i

m-2

am-i"based triangle a relation with the polynomial theorem. However, the structure of the latter triangle is much more complicated. See for example the triangle with abc base-number (Figure 5.) The elements in the n-th row are some sums of the coefficients of the polynomials (ax + by + cz)

n

. 1 abc a

2 2a b 2 ac+b

2 2 be c

2 a

3 2a

2 6 3a

2c+3aó

2 6abc+b

3 3ac

2+36

2c 36c

2 c

3 Figure 5: The abc-based triangle To discover the connection of the general triangle with the polynomial theorem we need the following. Definition. For the digits of the base-number let he weight of a digit be its distance from the centerline. So w(ao) = —w(a

m-i),w(ai) = —w(a

m2)) etc. If the base number is odd, then w(a^

m_iy

2) = 0. Let the unit of the weights be the distance of two neighbouring elements in the triangle, id est w(ai) = w(a

i

+1) = 1. Example. In the a&c-based triangle w(a) = —1, w(b) = 0 and u;(c) — 1, in the a&cd-based triangle w(a) — —1.5, w{b

) = —0.5, w(c) = 0.5 and w(d) = 1.5. We would like to extend this idea to the elements of the other rows. Because the elements of the triangle are sums, consider first the parts of them. For such an expression let the weight of the part be the sum of the weights of its digits. If a digit is on the z-th power then we count its weight i-times

. Example. One part of the third element in the third row of the abcbased triangle is 3a

2c (Figure 5.). For this expression we have g(3a

2c) = 2 w(a) + w(c) = -1. Lemma 1. In an element of the general triangle the weights of the parts are identical, and this weight is the distance of the element from the centerline.