On the properties of a sequence concerning binomial coefficients

TL;DR

This paper investigates the properties of a sequence defined by the minimal positive integer k where the binomial coefficient exceeds a specific exponential threshold, revealing new insights into binomial coefficient behavior.

Contribution

The paper introduces and analyzes the sequence f(n), providing new properties and understanding of binomial coefficients relative to exponential bounds.

Findings

Characterization of the sequence f(n) for n ≥ 3

Bounds and asymptotic behavior of f(n)

Relationships between binomial coefficients and exponential functions

Abstract

For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$.