are there any results on the number of graphs on n vertices that has chromatic number=k ?

I mean how many graphs on n vertices has x(G)=2,x(G)=3,......,x(G)=n-1 ?

**update**:
infact equivalently, is there a way to count unique k-partite graphs?
after that to get x(G)=k, just subtract #k-partite-#(k-1)-partite.
I've been almost able to count number with x(G)=2 using a much easier way but the problem remains for k>=3

This is an expression I got for k=2 but needs to be verified: $|\chi(G)=2|=n^{n-2}\frac{n+1}{2}=\frac{n^{n-1}+n^{n-2}}{2}$