0,3

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

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.

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

nn=22; 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+1, 2}]; Select[Range[0, nn]! CoefficientList[Series[Exp[cx-1], {x, 0, nn}], x], #>0&]

(PARI) seq(n)={my(g=log(sum(k=0, 2*n, 2^binomial(k, 2)*x^k/k!, O(x^(2*n+1)))), v=Vec(serlaplace(exp(sum(k=0, n, polcoef(g, 2*k)*x^(2*k), O(x^(2*n+1))))))); vector(n+1, i, v[2*i-1])} \\ Andrew Howroyd, Jan 20 2026

Cf. A001187, A218378, A182124.

Sequence in context: A198602 A214338 A084275 * A258488 A297052 A238566

Adjacent sequences: A218374 A218375 A218376 * A218378 A218379 A218380

Geoffrey Critzer, Oct 27 2012

a(10) corrected by Georg Fischer, Jan 20 2026

approved