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On the divisibility properties concerning sums of binomial coefficients

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Abstract

For any integer \(n> 1,\) we prove

$$\begin{aligned} 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(3k+1){2k\atopwithdelims ()k}^3(-8)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(6k+1){2k\atopwithdelims ()k}^3(-512)^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(42k+5){2k\atopwithdelims ()k}^3 4096^{n-1-k},\\ 2n{2n\atopwithdelims ()n}&\bigg |\sum _{k=0}^{n-1}(20k^2+8k+1){2k\atopwithdelims ()k}^5(-4096)^{n-1-k}. \end{aligned}$$

The first three results confirm three divisibility properties on sums of binomial coefficients conjectured by Z.-W. Sun.

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Acknowledgments

The author would like to thank the referee for his/her helpful comments.

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, Northwest A&F University, Yangling, 712100, Shaanxi, People’s Republic of China

    Bing He

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  1. Bing He
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Correspondence to Bing He.

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This work was supported by the Initial Foundation for Scientific Research of Northwest A&F University.

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He, B. On the divisibility properties concerning sums of binomial coefficients. Ramanujan J 43, 313–326 (2017). https://doi.org/10.1007/s11139-016-9780-6

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