Binomial coefficients have long been studied by several mathematicians for several centuries [1] and they are currently grouped in what is called Pascal's triangle [2]. These coefficients are very useful not only in combinatorics, but also, they intervene in many fields such as enumeration, development of the binomial in algebra, development in series, and in probability distributions and statistics [3]. In addition, Pascal's formula, generative of these coefficients, leads us to think of a generalization of these coefficients and of the other strongly related mathematical tools. In this article, we will try to generalize the binomial coefficients, the Newton’s binomial as well as the Fibonacci sequence and establish their expressions. We will call these generalizations p-nomial coefficients, Atta’s p-nomial and p-bonacci sequence.

p-nomial coefficients, p-nomial identity, staircase of p-nomials, p-bonacci sequence, p-Bernstein polynomial