On the inverse of a certain triangular matrix and its connection to the largest odd divisor

Sela Fried
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 255–262
DOI: 10.7546/nntdm.2026.32.2.255-262
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Authors and affiliations

Sela Fried
Department of Computer Science, Israel Academic College
52275 Ramat Gan, Israel

Abstract

The inverse R of a certain infinite triangular matrix A is shown to be directly related to the largest odd divisor function, thus proving a conjecture of Barry. We also provide a proof of a formula for R given by Yin and obtain bivariate generating functions for A and R.

Keywords

  • Bivariate generating function
  • Largest odd divisor
  • Lower triangular matrix

2020 Mathematics Subject Classification

  • 11A25
  • 05A15
  • 15A09

References

  1. Bencze, M. (2011). Problem 11553. The American Mathematical Monthly, 118(2), 178.
  2. Daiev, V. (1996). Problem H-81. The Fibonacci Quarterly, 4(1), 57.
  3. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics (2nd ed.). Addison–Wesley, Reading, MA.
  4. McKay, J. H. (1973). The William Lowell Putnam Mathematical Competition. The American Mathematical Monthly, 80(2), 170–179.
  5. Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation Inc. https://oeis.org.
  6. Wilf, H. S. (1994). Generatingfunctionology (2nd ed.). Academic Press. Available online at: https://www2.math.upenn.edu/~wilf/gfology2.pdf

Manuscript history

  • Received: 6 November 2025
  • Revised: 20 March 2026
  • Accepted: 30 March 2026
  • Online First: 1 April 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Fried, S. (2026). On the inverse of a certain triangular matrix and its connection to the largest odd divisor. Notes on Number Theory and Discrete Mathematics, 32(2), 255-262, DOI: 10.7546/nntdm.2026.32.2.255-262.

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