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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 (list; graph; refs; listen; history; text; internal format)
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^(m-n+c) where m,n,c is the number of edges, vertices, and components of G respectively. - Geoffrey Critzer, Feb 23 2020

LINKS

Table of n, a(n) for n=0..10.

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 o-o o-o.

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

CROSSREFS

Cf. A001187, A218378, A182124.

Sequence in context: A198602 A214338 A084275 * A258488 A297052 A238566

Adjacent sequences:  A218374 A218375 A218376 * A218378 A218379 A218380

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Oct 27 2012

STATUS

approved

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Last modified October 23 17:37 EDT 2021. Contains 348215 sequences. (Running on oeis4.)