Representation of Power Functions in Mathematical Series

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m ≥ 0 as sum of row terms of corresponding triangle Structure of the manuscript The problem of finding expansions of monomials, binomials, trinomials, etc. is classical and a lot of theorems have been found, the most prominent examples are Binomial Theorem [2], Multinomial theorem, Wozpitsky Identity [30], Stirling numbers of second kind identity, etc. In this paper we try to solve the classical problem of finding expansions of monomials. We start from binomial distributed triangle A287326 [11] in OEIS. The main property of A287326 that it returns a perfect cube n as n-th row sum, starting from 0,...,n − 1 or from 1,...,n by means of its symmetry. Therefore, the following question stated: • Can we find similar to A287326 triangles in order to receive monomial nt, t > 3 as sum of row terms? In other words, can A287326 be generalized in order to receive monomial nt, t > 3 as sum of row terms?