According to O'Donnell's book ``Analysis of Boolean Functions", in order to determine the Fourier coefficient of a boolean function $f$ on a subset $S$, we take an inner product of $\chi_S$ and $f$ and then divide the result by $2^n$. In the paper by Beigel [1], a different formula is given. Specifically, $c_{S}^{fourier}(f)=-2^{-n-1}\Sigma_{T\subseteq[n]}(1-2c_T^{table}(f))(1-2(|T\cap S|\mod 2))$. I understand how the sum corresponds to the inner product in the formula from the book, but I cannot figure out why the result is divided by $-2^{n+1}$.
[1] Beigel, Richard. "The Polynomial Method in Circuit Complexity."
