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 17 11 1 + — : : + •••+(TO - 1)! A'o + i\ 4 h *m-l \ (i

0-l)!(zi -l)!-..(i

m_i -1)! V »0*1 •••»m-1 / By these three Lemmas we have proved the following. Theorem 2. The elements in the n-th row of the 00^1^2 * • • im-2

öm-i based triangle are exactly the sums of the coefficients of the polynomial (a

0x

0 + aixi +azx

2 H I a + a

m_ix

m_i)

n, in which the weights of the parts are identical. Like among the binomial coefficients in Pascal's triangle (for example Edwards [1] and Vilenkin [2]), in the general triangle there are also interesting connecitons among the elements. One of them comes immediately from the second Theorem. Corollary. In the n-th row of the ao^i^ • • • -2^m-i -based triangle the sum of the elements (with normal addition) is (ao -f a\ -f 02 + • • • + am-2 + «m-i )

n • Proof. If we set in the polynomial (a

0£o + a\X\ + a

2x

2 + b a

m_

2

xm2 + a

m-i

xm-i)

n, 1 = xq = xi = X2 — ••• — x

m-2 — xm-i> then from Theorem 2 in the TO-th row of the triangle there are the coefficients of the "polynomial" (a

0 + fli + a

2 + H a

m2 + fl

m-1)

2Remark. In Pascal's triangle from this Corollary we get the well known combinatorical equality Another possibiHty to power polynomials is that we extend the property for the general triangle, that the elements in the TO-th row of Pascal's Triangle are the coefficients of the binomial 1 + x. Proposition 4. The elements in the n-th row of the general triangle are exactly the coefficients of the polynomials (ao + ßi x + a^x} -f • • + a

m_

2x

m~

2 -f a

m_ix

m1)

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

k

. Proof. We prove by induction. In the first row the statement is true. Let us now assume, that in the TO — 1-th row there are the coefficients of the . W ;„UAKGKR T j Könyv: QC2 , CtQ{