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%I #15 Feb 24 2020 04:36:44
%S 1,1,41,27289,252354929,34508040597841,73356878424474928601,
%T 2471655487735117774297253929,1328579254939122192980041623517564769,
%U 11416413723707413064765254593047001003783424801,2169118832800743175599952429700612077287847317513
%N Number of simple labeled graphs on 2n nodes with all even size components.
%C These are precisely the graphs G in which there exists a spanning subgraph F of G such that every vertex in F has odd degree. The number of such subgraphs in any such graph G is 2^(m-n+c) where m,n,c is the number of edges, vertices, and components of G respectively. - _Geoffrey Critzer_, Feb 23 2020
%F E.g.f. for the sequence with interpolated 0's is: exp( ( A(x) + A(-x) - 2 )/2) where A(x) is the e.g.f. for A001187.
%e a(2) = 41 because (on 4 labeled nodes) we have 38 connected graphs and 3 in the isometry class o-o o-o.
%t nn=20; a=Sum[2^Binomial[n,2]x^n/n!, {n,0,nn}]; c=Range[0,nn]! CoefficientList[Series[ Log[a]+1, {x,0,nn}], x]; cx= Sum[c[[i]]x^(i-1)/(i-1)!, {i,1,nn,2}]; Select[Range[0,nn]! CoefficientList[Series[Exp[cx-1], {x,0,nn}], x], #>0&]
%Y Cf. A001187, A218378, A182124.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Oct 27 2012
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