The generalization of Pascal’s triangle from algebraic point of view

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 French scientist, but was known already by the ancient Chinese and others before Pascal (Edwards [1]). 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 non-negativ integer, and the indices in the rows and columns run from 0. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 · · · 1 = 11, 11 = 11, 121 = 11, 1331 = 11, 14641 = 11, 161051 = 11, . . . Figure 1: The powers of 11 in Pascal’s triangle This comes immediately from the binomial equality