Products and sums divisible by central binomial coefficients

2010/04/26 by Zhi-Wei Sun, Zhi‐Wei Sun, Sun, Zhi-Wei
Mathematics · #05A10 #11A07 #11B65 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:11A07 #msc:11B65

paper · pdf · doi:10.48550/arxiv.1004.4623

15 pages. Submitted version. Add some conjectures and references.

openalex publication_date 2010/04/26 · arxiv created 2010/05/05 · arxiv updated 2010/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers k and n we have \binom 2kk | \binom4n+2k+22n+k+1\binom2n+k+12k\binom2n-k+1n and \binom2kk | (2n+1)\binom2nnC_n+k\binomn+k+12k, where C_m denotes the Catalan number \binom2mm/(m+1)=\binom2mm-\binom2mm+1. Applying this result we obtain two sums divisible by central binomial coefficients.

Related

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