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 13 Proposition 3. Connection with the binomial theorem. The elements in the n-th row of the ab-based triangle are the coefficients of the polynomials (ax -f by)

n

. Proof. If we substitute ax with 10a and by with b in the Proof of Proposition 1, and use that the k-th element in the n-th row of the abbased triangle is a

n~

kb

kC

k we get the statement. Example. From the 47-based triangle (4x + 7y)

3 = 64a;

3 -f 336x

2y + 588xy

2 + 343

y

3

. The base-number of the triangle can consist of not only 2, but arbitrarily many digits. Definition. Let 0 < ao, ai, • • • ? am-2, ßm-i < 9 be integers. Then we can get the k-th element in the n-th row of the ao

ßi

a2 • ••a

m-2

(

lm-r based triangle if we multiply the k — ra-th element in the n — 1-th row by a

m-1, the k — m -f 1-th element in the n — 1-th row by ao, and add the products. If for some i we have k — m + i < 0 or k — m + i > n — 1 (id est some element in the n — 1-th row does not exists according to the traditional implementation) then we consider this element to be 0. The indices in the rows and columns of the triangle run from 0 (Figure 3.). 1 4 3 5 16 24 49 30 25 64 144 348 387 435 225 125 Figure 3: The 435-based triangle Remarks. In the above definition we can allow for the base-number not only 0 < ao, a\

, a2,

..., a

m_2

, a

m_i < 9 digits, but arbitrary integers, rational and irrational numbers. Thus for example we can build triangles with base of root expressions (Figure 4.). l V2 V3 V5 2 2\/6 2VTÖ+3 2v

/15 5 2V2 6^/3 6V5+9V3Ö 6v

/3Ö+3\/3 9n/5+15 v"2 15N/3 5^5 Figure 4: The \/2 \/3 y/b-based triangle