Products and Sums Divisible by Central Binomial Coefficients

Abstract

In this paper we study products and sums divisible by central binomial coefficients. We show that $$2(2n+1)\binom{2n}n\ \bigg|\ \binom{6n}{3n}\binom{3n}n\ \ \mbox{for all}\ n=1,2,3,\ldots.$$ Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k\ \bigg|\ \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k\ \bigg|\ (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k},$$ where $C_m$ denotes the Catalan number $\frac1{m+1}\binom{2m}m=\binom{2m}m-\binom{2m}{m+1}$. On the basis of these results, we obtain certain sums divisible by central binomial coefficients.