Bisecting binomial coefficients

TL;DR

This paper explores the problem of bisecting binomial coefficients, discovering new classes of integers with nontrivial bisections, proving related conjectures, and providing bounds and exact counts for small cases.

Contribution

It introduces new infinite classes of integers with nontrivial bisections, proves conjectures Q2 and Q4, and computes exact counts for n ≤ 51.

Findings

Identified many new classes of integers with nontrivial bisections.

Proved conjectures Q2 and Q4 of Cusick and Li.

Computed the exact number of bisections for n ≤ 51.

Abstract

In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n <= 51.