The Series of Reciprocals of Non-central Binomial Coefficients


Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients. As the result of splitting items, some identities of series of numbers values of reciprocals of binomial coefficients are given. The method of splitting terms offered in this paper is a new combinatorial analysis way and elementary method to create new series.

Share and Cite:

L. Zhang and W. Ji, "The Series of Reciprocals of Non-central Binomial Coefficients," American Journal of Computational Mathematics, Vol. 3 No. 3B, 2013, pp. 31-37. doi: 10.4236/ajcm.2013.33B006.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] B. Sury, T. N. Wang and F. Z. Zhao, “Some Identities Involving of Binomial Coefficients,” J. integer Sequences, Vol. 7, 2004,Article 04.2.8
[2] J. H. Yang and F. Z. Zhao, “Sums Involving the Inverses of Binomial Coefficients, Journal of Integer Sequences, Vol. 9, 2006, Article 06.4.2
[3] S. Amghibech, “On Sum Involving Binomial Coefficient,” Journal of Integer Sequences, Vol.10, 2007, Article 07.2.1
[4] T. Trif, “Combinatorial Sums and Series Involving Inverses of Binomial Coefficients,” Fibonacci Quarterly, Vol. 38, No. 1, 2000, pp. 79-84.
[5] F.-Z Zhao and T. Wang, “Some Results for Sums of the Inverses of Binomial Coefficients,” Integers: Electronic, Journal of combinatorial Number Theory, Vol. 5, No. 1, 2005, p. A22.
[6] R. Sprugnoli, “Sums of Reciprocals of the Central Binomial Coefficients,” Integers: Electronic Journal of Combinatorial Number Theory, Vol. 6 ,2006, p. A27
[7] I. S. Gradshteyn and I. M. Zyzhik, “A Table of Integral, Series and Products,” Academic Press is an Imprint of Elsevier, Seventh Edition, Vol. 56, No. 61.
[8] W. H. Ji and L. P. Zhang, “On Series Alternated with Positive and Negative Involving Reciprocals of Binominal Coefficients,” Pure Mathematics, Vol. 2, No. 4, 2012, pp. 192-201. doi:10.4236/PM.2012.24030
[9] W. H. Ji and B. L. Hei, “The Series of Reciprocals of Binomial Coefficients Constructing by Splitting Terms,” Pure Mathematics, 2013, Vol. 3, No. 1, p. 18.doi:10.12677/PM.2013.31005

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.