Abstract
In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize the relationship between the new triangles and the Pascal pyramids, moreover we present connections with the binomial and multinomial theorems.
Highlights
The interesting and really romantic Pascal triangle has been a favorite research field for mathematicians for a very long time
Later in Persia, China and Europe in the Middle Ages by a number of scientists before Pascal, he generalized known results and gave a number of new properties, which he formulated in nineteen theorems
The so-called Pascal pyramid is constructed from trinomial coefficients, which occur in the expansions (x + y + z)n
Summary
The interesting and really romantic Pascal triangle has been a favorite research field for mathematicians for a very long time. Later in Persia, China and Europe in the Middle Ages by a number of scientists before Pascal, he generalized known results and gave a number of new properties, which he formulated in nineteen theorems ([3]). We can construct the generalized Pascal triangles of sth order (or kind s, sometimes referred to as the s-arithmetical triangles), from the generalized binomial coefficients of order s. This idea was first published in 1956 by. These triangles have been intensively investigated in the last decades, we cite some important properties in Sections 4 and. Other interesting planar generalizations are the Lucas, Fibonacci, Catalan and other arithmetical triangles, these constructions diverge from our topic.
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