The integrality of the Genocchi numbers obtained through a new identity and other results

Bakir Farhi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 4, Pages 749–757
DOI: 10.7546/nntdm.2022.28.4.749-757
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Authors and affiliations

Bakir Farhi
Laboratoire de Mathématiques appliqueés,
Faculté des Sciences Exactes, Université de Bejaia
06000 Bejaia, Algeria

Abstract

In this note, we investigate some properties of the integer sequence of general term a_n := \sum_{k = 0}^{n - 1} k! (n - k - 1)! (\forall n \geq 1) to derive a new identity of the Genocchi numbers G_n (n \in \mathbb{N}), which immediately shows that G_n \in \mathbb{Z} for any n \in \mathbb{N}. In another direction, we obtain nontrivial lower bounds for the 2-adic valuations of the rational numbers \sum_{k = 1}^{n} \frac{2^k}{k}.

Keywords

  • Genocchi numbers
  • Stirling numbers
  • Binomial coefficients
  • p-adic valuations

2020 Mathematics Subject Classification

  • 11B65
  • 11B68
  • 11B73

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Manuscript history

  • Received: 21 May 2022
  • Revised: 14 October 2022
  • Accepted: 12 November 2022
  • Online First: 14 November 2022

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Cite this paper

Farhi, B. (2022). The integrality of the Genocchi numbers obtained through a new identity and other results. Notes on Number Theory and Discrete Mathematics, 28(4), 749-757, DOI: 10.7546/nntdm.2022.28.4.749-757.

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