A218377 Number of simple labeled graphs on 2n nodes with all even size components.

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

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