Correlation of an approximation over a random permutation
So $x$ measures the departure from balance in the left half of the truth table, say it is the excess number of 1's in the left half of length $2^{n-1}$. By the definition of binomial coefficients there are $$\binom{2^{n-1}}{2^{n-2}+x}$$ such vectors. This is the first factor. The second factor is the count of vectors missing $x$ 1's, since the overall truth table is balanced.

Using $$\binom{a}{b}=\binom{a}{a-b}$$ completes the argument.