On an upper bound for central binomial coefficients and Catalan numbers

TL;DR

This paper compares different upper bounds for central binomial coefficients and Catalan numbers, showing that a recent bound is looser than a classical one, and introduces improved bounds that closely approximate the exact values.

Contribution

The paper demonstrates that Agievich's recent upper bound exceeds Sasvari's classical bound for central binomial coefficients, and derives next-order bounds with improved accuracy for Catalan numbers.

Findings

Agievich's bound is larger than Sasvari's for central binomial coefficients.

Next-order bounds for Catalan numbers are very close to the exact values.

The difference between bounds and exact values decreases with larger n and higher order.

Abstract

Recently, Agievich proposed an interesting upper bound on binomial coefficients in the de Moivre-Laplace form. In this article, we show that the latter bound, in the specific case of a central binomial coefficient, is larger than the one proposed by Sasvari and obtained using the Binet formula for the Gamma function. In addition, we provide the expression of the next-order bound and apply it to Catalan numbers $C_n$. The bounds are very close to the exact value, the difference decreasing with $n$ and with the order of the upper bound.