Difference of two binomial random variables with identical distribution

Question

Is there a closed form answer for absolute value of difference of two identical binomial random variables with identical distributions when $p=1/2$? In particular, what is the the distribution of $|X-Y|$ when $X$ and $Y$ are independent and both $~Bin(n,1/2)$?

Answer 1

From the various answers at the linked question, it looks as if $$P(|X-Y|=0)={2n \choose n} \frac{1}{2^{2n}}$$ while for positive $z$ $$P(|X-Y|=z)={2n \choose n+z} \frac{1}{2^{2n-1}}$$