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%I #6 Mar 30 2012 17:26:52
%S 0,7,640,170555,94949400,95830621425,159062872168200,
%T 404720953797785625
%N a(n) = number of labeled 3-regular (trivalent) multi-graphs without self-loops on 2n vertices with a maximum of 2 edges between any pair of nodes. Also a(n) = number of labeled symmetric 2n X 2n matrices with {0,1,2}-entries with row sum equal to 3 for each row and trace 0.
%e a(2)=7 because for 2*n=4 nodes there are 7 possible labeled graphs whose adjacency matrices are as follows:
%e 0 2 1 0
%e 2 0 0 1
%e 1 0 0 2
%e 0 1 2 0;
%e 0 1 2 0
%e 1 0 0 2
%e 2 0 0 1
%e 0 2 1 0;
%e 0 2 0 1
%e 2 0 1 0
%e 0 1 0 2
%e 1 0 2 0;
%e 0 1 1 1
%e 1 0 1 1
%e 1 1 0 1
%e 1 1 1 0;
%e 0 0 2 1
%e 0 0 1 2
%e 2 1 0 0
%e 1 2 0 0;
%e 0 1 0 2
%e 1 0 2 0
%e 0 2 0 1
%e 2 0 1 0;
%e 0 0 1 2
%e 0 0 2 1
%e 1 2 0 0
%e 2 1 0 0.
%Y Cf. A001205, A002829, A108243.
%K nonn,more
%O 1,2
%A _Jeremy Gardiner_, Aug 29 2005
%E a(5)-a(8) from _Max Alekseyev_, Aug 30 2005
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