Products and sums divisible by central binomial coefficients

Zhi-Wei Sun

paper

Products and sums divisible by central binomial coefficients

arXiv:1004.4623

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

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

Cited by in corpus (1)

On divisibility concerning binomial coefficients