Binomial Coefficients and Triangular Numbers Binomial coefficients and triangular numbers



Abstract Views
339

Downloads
518

Citations

Share

##plugins.themes.bootstrap3.article.sidebar##

Submitted Mar 16, 2021
Published Jul 5, 2021
10.24018/ejmath.2021.2.3.31

Abstract viewed = 339 times

Table of Contents

##plugins.themes.bootstrap3.article.main##

We produce formulas of sums the product of the binomial coefficients and triangular numbers. And we apply our formula to prove an identity of Wang and Zhang. Further, we provide an analogue of our identity for the alternating sums.

Keywords: binomial coefficient; triangular number

References

  1. N. J. Calkin, “A curious binomial identities,” Discrete Mathematics, vol. 131, pp. 335-337, 1994.
     Google Scholar
  2. L. Depnarth, “A short history of the Fibonacci and golden numbers with their applications,” Math. Educ. Sci. Technol., vol. 42, pp. 337-367, 2011.
     Google Scholar
  3. T. Koshy, Fibonacci and Lucas numbers with applications, New York, NY: John Wiley and Sons; 2001.
     Google Scholar
  4. V. E. Hoggatt, Jr., “Some special Fibonacci and Lucas generating functions,” The Fibonacci Quarterly, vol. 19, no. 2, pp. 121-138, 1971.
     Google Scholar
  5. M. Hirschhorn, “Calkin’s binomial identity,” Discrete Mathematics, vol. 159, pp. 273-278, 1996.
     Google Scholar
  6. J. V. Leyendekkers and A. G. Shannon, “Integer structure analysis of odd powered triples: The significance of triangular versus pentagonal number,” Notes on Number Theory and Discrete Mathematics, vol. 16, no. 4, pp. 6-13, 2010.
     Google Scholar
  7. N. J. A. Sloane, A Handbook of Integer Sequences, San Diego, CA: Academic Press, 1964. [The On-Line Encyclopedia of Integer Sequences oeis.org (OEIS)].
     Google Scholar
  8. J. Wang and Z. Zhang, “On extensions of Calkin's binomial identities,” Discrete Mathematics, vol. 274, pp. 331-342, 2004.
     Google Scholar