Solvability of $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$

TL;DR

This paper characterizes the conditions under which the binomial coefficient equation 2k choose k = 2a choose a  x+2b choose b holds, proving it only when x=a=1, using various proof techniques.

Contribution

The paper provides a complete characterization of the solvability of a specific binomial coefficient equation, establishing that it holds only for x=a=1.

Findings

The equation holds if and only if x=a=1.

Multiple proof techniques are employed, including direct proof and computational verification.

Previous related results are incorporated to support the proof.

Abstract

Suppose $k,x,$ and $b$ are positive integers, and $a$ is a nonnegative integer such that $k=a+b$. In this paper, we will prove $\binom{2k}{k} = \binom{2a}{a} \binom{x+2b}{b}$ if and only if $x=a=1$. We do this by looking at different cases depending on the values of $x$ and $k$. We use varying techniques to prove the cases, such as direct proof, verification through Maple software, and a proof technique found in Moser's paper. Previous results from Hanson, St\u{a}nic\u{a}, Shanta, Shorey and Nair are also used.