Number of symmetric relations possible= $2^n.2^{\frac{n. (n-1)}{2}}$
Out of which those relations that contain only reflexive pairs can be symmetric and antisymmetric at the same time. $2^n$ such relations are possible.
So the number of relations which are symmetric but not antisymmetric = $2^n.2^{\frac{n. (n-1)}{2}} - 2^n = 2^n(2^{\frac{n. (n-1)}{2}}-1)$