There is unlikely to be a formula as such, although as noted by others it can be computed from the values for not necessarily connected graphs. (See A263293 in OEIS for values up to 10 nodes - these values were obtained by brute force enumeration of all graphs). Better methods exist, since it isn't necessary to enumerate every graph - only every degree sequence, but to my knowledge this has not been done. This technique, might give values up to about 15 nodes, but it is still exponential.