Uniform balanced binary sequence sampling

A balanced binary sequence of length $2n$ has $n$ entries of each kind. Uniformly sample an $n$-element subset of the positions and place the first kind there. If $2n$ is a power of two, repeatedly draw uniform positions using ${\log }_2(2n)$ fair random bits, reject previously selected positions, and stop after $n$ distinct positions. Each new position is uniform among those remaining, making every $n$-element subset equally likely. The expected number of draws is ${\sum }_{s=0}^{n-1}2n/(2n-s)<2n$, giving $O(n\log n)$ expected random bits.