Preprint Article Version 6 This version is not peer-reviewed

Series Representation of Power Function

Version 1 : Received: 24 November 2017 / Approved: 24 November 2017 / Online: 24 November 2017 (05:15:26 CET)
Version 2 : Received: 12 January 2018 / Approved: 18 January 2018 / Online: 18 January 2018 (03:37:52 CET)
Version 3 : Received: 18 February 2018 / Approved: 19 February 2018 / Online: 19 February 2018 (16:42:41 CET)
Version 4 : Received: 8 May 2018 / Approved: 9 May 2018 / Online: 9 May 2018 (06:31:05 CEST)
Version 5 : Received: 28 May 2018 / Approved: 28 May 2018 / Online: 28 May 2018 (08:26:14 CEST)
Version 6 : Received: 16 August 2018 / Approved: 17 August 2018 / Online: 17 August 2018 (11:10:47 CEST)

How to cite: Kolosov, P. Series Representation of Power Function. Preprints 2017, 2017110157 (doi: 10.20944/preprints201711.0157.v6). Kolosov, P. Series Representation of Power Function. Preprints 2017, 2017110157 (doi: 10.20944/preprints201711.0157.v6).

Abstract

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. In other words, in this manuscript are found and discussed the polynomials Dm(n,k) and Um(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n2m+1.

Subject Areas

series representation; power function; monomial; binomial theorem; multinomial theorem; worpitzky identity; stirling numbers of second kind; faulhaber's sum; finite difference; faulhaber's formula; central factorial numbers; binomial coefficients; binomial distribution; binomial transform; bernoulli numbers; oeis; multinomial coefficients

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