Expression for binomial coefficient denominator

Question

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 1

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

You can find several formulas and details there.

Answer 2

We have $$ \binom{-1/2}{k} = (-1)^k \frac{(1)(3)(5)\cdots(2k-1)}{2^k k!} = (-1)^k \frac{(2k)!/(2^k k!)}{2^k k!} = (-1)^k \frac{1}{2^{2k}} \binom{2k}{k}. $$ A result of Kummer states that the highest power of 2 dividing $\binom{2k}{k}$ is the number of carries when adding $n$ to itself in base 2, which in this case is easily seen to be the number of 1s in the binary representation of $k$. Hence the answer is $2^{2k-|k|_2}$, where $|k|_2$ is the number of 1s in the binary representation of $k$.