On some series involving the binomial coefficients $\binom{3n}{n}$

TL;DR

This paper simplifies and evaluates series involving binomial coefficients 3n, explores their connections with Fibonacci numbers and the golden ratio, and introduces new Fibonacci and Lucas series with reciprocals of these coefficients.

Contribution

It provides simpler forms and new evaluations of series involving 3n, establishing novel links with Fibonacci numbers and the golden ratio.

Findings

Simplified forms for series involving 3n coefficients.

New evaluations connecting binomial series with Fibonacci numbers.

Derived Fibonacci and Lucas series involving reciprocals of 3n.

Abstract

Using a simple transformation, we obtain much simpler forms for some series involving binomial coefficients $\binom{3n}n$ derived by Necdet Batir. New evaluations are given; and connections with Fibonacci numbers and the golden ratio are established. Finally, we derive some Fibonacci and Lucas series involving the reciprocals of $\binom{3n}{n}$.