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

18 Gábor Kallós polynomial (a

0 + o,\x + a

2x

2 + 1- a

m2x

m~

2 + a

m_ix

m~

l )

n_

1, the k-th element is the coefficient of x

k. If we multiply this polynomial by do + d\X + a

2x

2 + • • • + a

m2x

m~

2 -f a

m_i®

m1, and add up the results (similarly as in the proof of Theorem 1), we get the n-th power of the basic polynomial. But according to the forming rules of the triangle, the coefficients of this polynomial are exactly the elements of the n-th row. Example. Prom the third row of the 435-based triangle (Figure 3.) (4 + 3a: + 5x

2)

3 = 64 + 144a: + 348a:

2 + 387a:

3 + 435a:

4 + 225a:

5 + 125a:

6. Consideration of effectivity. The powering of polynomials is considerably more complex operation as powering of (integer) numbers. However, the consideration above applies here, too. So if we need only the n-th power of the base-polynomial some other methods are more effective (Knuth [4], Geddes [6].) However, if we need all the (non-negativ integer) powers up to n then this method is competitive. References [1] EDWARDS, A. W. F. Pascal's Arithmetical Triangle, Charles Griffin and Company Ltd, Oxford University Press, 1987. [2] VlLENKIN, N. J.: Kombinatorika. Tankönyvkiadó, Budapest, 1987. [3] GEROCS L.: A Fibonacci-sorozat általánosítása. Tankönyvkiadó, Budapest, 1988. [4] KNUTH, D. E., The Art of Computer Programming, Vol. 2. Seminumerical Algoritms, Addison-Wesley, 1981. [5] SURANYI, J.: Számoljunk ügyesen. KÖMAL XXV. kötet, 1962. [6] GEDDES, K. O., CZAPOR, S. R., LABAHN, G., Algorithms for Computer Algebra, Kluwer Academic Publishers, 1991. SZÉCHENYI ISTVÁN COLLÉGE HÉDERVÁRI u. 3. H-9026 GYŐR, HUNGARY E-mail: kallos@rsl.szif.hu