Bisecting binomial coefficients

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 (2005). 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.