Couting Self dual functions
Using de Morgan's laws, we get a more useful characterization of self-dual functions: those are functions satisfying $\lnot F(x_1,\ldots,x_n) = F(\lnot x_1,\ldots,\lnot x_n)$. Suppose that you knew the value of a self-dual function on all inputs in which $x_1$ is false. Can you recover the rest of $F$? Can you deduce the number of self-dual functions?