Products and sums divisible by central binomial coefficients

TL;DR

This paper explores divisibility properties of products and sums involving central binomial coefficients, establishing new divisibility formulas and deriving sums that are divisible by these coefficients.

Contribution

It introduces novel divisibility results for products and sums related to central binomial coefficients, expanding understanding of their algebraic properties.

Findings

Proves divisibility of specific binomial products for all positive integers.

Derives sums involving binomial coefficients that are divisible by central binomial coefficients.

Provides new formulas connecting binomial coefficients and Catalan numbers.

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 $$\binom {2k}k | \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k | (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k},$$ where $C_m$ denotes the Catalan number $\binom{2m}m/(m+1)=\binom{2m}m-\binom{2m}{m+1}$. Applying this result we obtain two sums divisible by central binomial coefficients.