

A218377


Number of simple labeled graphs on 2n nodes with all even size components.


1



1, 1, 41, 27289, 252354929, 34508040597841, 73356878424474928601, 2471655487735117774297253929, 1328579254939122192980041623517564769, 11416413723707413064765254593047001003783424801, 2169118832800743175599952429700612077287847317513
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OFFSET

0,3


COMMENTS

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^(mn+c) where m,n,c is the number of edges, vertices, and components of G respectively.  Geoffrey Critzer, Feb 23 2020


LINKS



FORMULA

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.


EXAMPLE

a(2) = 41 because (on 4 labeled nodes) we have 38 connected graphs and 3 in the isometry class oo oo.


MATHEMATICA

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^(i1)/(i1)!, {i, 1, nn, 2}]; Select[Range[0, nn]! CoefficientList[Series[Exp[cx1], {x, 0, nn}], x], #>0&]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



