A182012 - OEIS

%I #49 Jan 02 2023 12:30:48

%S 1,3,16,243,33120,87723296,3633057074584,1967881448329407496,

%T 13670271807937483065795200,1232069666043220685614640133362240,

%U 1464616584892951614637834432454928487321792,23331378450474334173960358458324497256118170821672192,5051222500253499871627935174024445724071241027782979567759187712

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

%C As usual, "graph" means "simple graph, without self-loops or multiple edges".

%C 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 Sequence Fans Mailing List, <a href="http://list.seqfan.eu/oldermail/seqfan/2012-April/016735.html">Discussion</a>, April 2012.

%H N. J. A. Sloane, <a href="/A182012/a182012.pdf">The 16 graphs on 6 nodes</a>

%F a(n) = A002854(2n).

%e 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 (Sage)

%o def graphsodddegree(MAXN=5):

%o     """

%o     requires optional package "nauty"

%o     """

%o     an=[]

%o     for n in range(1,MAXN+1):

%o         gn=graphs.nauty_geng("%s"%(2*n))

%o         cac={}

%o         a=0

%o         for G in gn:

%o             d = G.degree_sequence()

%o             if all(i % 2 for i in d):

%o                 a += 1

%o         print('a(%s)=%s'%(n,a))

%o         an += [a]

%o     return an

%Y Cf. A210345, A210346, A000088. Bisection of A002854.

%K nonn,easy

%O 1,2

%A _Georgi Guninski_, Apr 06 2012

%E Terms from a(6) on added from A002854. - _N. J. A. Sloane_, Apr 08 2012