How many self-dual Boolean functions satisfy specific conditions?

[mathwo]

A Boolean function of n variables is a mapping f : {0, 1}^n to {0, 1}. . Determine the number of Boolean functions f of n variables such that
(i) f is not self-dual and f(0, 0, . . . , 0) = f(1, 1, . . . , 1),
(ii) f is self-dual and f(0, 0, . . . , 0) = 1.
I think for the first part, i need to find the number of functions that is not self-dual, then find the number of functions i need from it? For the second part,i absolutely have no clue, please help me with this question.

 

[mfb]

For the first part, I would calculate the number of self-dual functions, and the number of functions where f(0, 0, . . . , 0) = f(1, 1, . . . , 1). As all self-dual functions satisfy that condition, both numbers are sufficient to get the answer.

For the second part: Well, you know f(1, 1, . . . , 1) as well. How many independent function values are left which you can choose?