For n=2,4,6,...,30 the number of graphical partitions is 1, 2, 5, 9, 17, 31, 54, 90, 151, 244, 387, 607, 933, 1420, 2136. This is sequence Anum=A000569"> A000569 in
The program that produces this output is derived from one kindly supplied to us by Carla Savage. Her program is based on an algorithm described in the paper: T.M. Barnes and C.D. Savage, Efficient Generation of Graphical Partitions, submitted manuscript, 1995. See also the paper: T.M. Barnes and C.D. Savage, A recurrence for counting graphical partitions, Electronic Journal of Combinatorics, 2, 1995, #R11.
Further information can be obtained from Eric Weinstein's World of Mathematics.