Article
Version 5
Preserved in Portico This version is not peer-reviewed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)
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. https://doi.org/10.20944/preprints201711.0157.v5 Kolosov, P. Series Representation of Power Function. Preprints 2017, 2017110157. https://doi.org/10.20944/preprints201711.0157.v5
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.
Supplementary and Associated Material
https://oeis.org/A287326: Sequence A287326 in OEIS, triangle, returns n^3 as sum of n-th row terms, where n is natural.
https://oeis.org/A300656: Sequence A300656 in OEIS, triangle, returns n^5 as sum of n-th row terms, where n is natural.
https://oeis.org/A300785: Sequence A300785 in OEIS, triangle, returns n^7 as sum of n-th row terms, where n is natural.
https://oeis.org/A302971: Sequence, used in generalization of A287326, A300656, A300785 for every odd power
Keywords
power function; monomial; binomial coeffcient; binomial theorem; finite difference; perfect cube; Pascal's triangle; series representation; Faulhaber's formula
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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