fractions - Expression for binomial coefficient denominator

I'm trying to find an analytical expression for the denominator of $\pmatrix{-1/2\\k}$ in terms of $k$ when the fraction is fully reduced.

E.g., the first several such denominators, starting with $k=0$, are $1,2,8,16,128,256,1024,2048,32768$, so there are various power-of-$2$ jumps, but I haven't been able to figure out the overall pattern so that I can nail down the expression.

Does anyone know of such an expression, or know of a good place to look to try to figure this out? If not, does anyone know if this is a fool's errand?

Thanks for any help.

Answer

This is integer sequence A046161 in the On-Line Encyclopedia of Integer Sequences.

You can find several formulas and details there.