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A182012 Number of graphs on 2n unlabeled nodes all having odd degree.

%I A182012 #50 Dec 23 2024 14:53:42
%S A182012 1,3,16,243,33120,87723296,3633057074584,1967881448329407496,
%T A182012 13670271807937483065795200,1232069666043220685614640133362240,
%U A182012 1464616584892951614637834432454928487321792,23331378450474334173960358458324497256118170821672192,5051222500253499871627935174024445724071241027782979567759187712
%N A182012 Number of graphs on 2n unlabeled nodes all having odd degree.
%C A182012 As usual, "graph" means "simple graph, without self-loops or multiple edges".
%C A182012 The graphs on 2n vertices all having odd degrees are just the complements of those having all even degrees. That's why the property of all odd degrees is seldom mentioned. Therefore this sequence is just every second term of A002854. - _Brendan McKay_, Apr 08 2012
%H A182012 Sequence Fans Mailing List, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2012-April/016735.html">Discussion</a>, April 2012.
%H A182012 N. J. A. Sloane, <a href="/A182012/a182012.pdf">The 16 graphs on 6 nodes</a>
%F A182012 a(n) = A002854(2n).
%e A182012 The 3 graphs on 4 vertices are [(0, 3), (1, 3), (2, 3)], [(0, 2), (1, 3)], [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]: the tree with root of order 3, the disconnected graph consisting of two complete 2-vertex graphs, and the complete graph.
%o A182012 (Sage)
%o A182012 def graphsodddegree(MAXN=5):
%o A182012     """
%o A182012     requires optional package "nauty"
%o A182012     """
%o A182012     an=[]
%o A182012     for n in range(1,MAXN+1):
%o A182012         gn=graphs.nauty_geng("%s"%(2*n))
%o A182012         cac={}
%o A182012         a=0
%o A182012         for G in gn:
%o A182012             d = G.degree_sequence()
%o A182012             if all(i % 2 for i in d):
%o A182012                 a += 1
%o A182012         print('a(%s)=%s'%(n,a))
%o A182012         an += [a]
%o A182012     return an
%Y A182012 Cf. A210345, A210346, A000088. Bisection of A002854.
%K A182012 nonn,easy
%O A182012 1,2
%A A182012 _Georgi Guninski_, Apr 06 2012
%E A182012 Terms from a(6) on added from A002854. - _N. J. A. Sloane_, Apr 08 2012