I would like to compute the Tutte polynomial of the complete graph $K_n$ for n as large as possible. Using a program by Björklund, Husfeldt, Kaski, Koivisto (here), I managed to compute up to n=18 on my home computer (in serial) in less than a day. Overall, I've been very impressed by this program.
I'm likely to include the results of these computations in an upcoming paper, so, for comparison, I would also like to mention what people have done previously in the area. (Also, I'd like to check that the computations are consistent with one another.)
Question: How far have others computed the Tutte polynomial of the complete graph?