A274934 Number of unlabeled graphs with n nodes that have two components, neither of which is the empty graph.

0, 0, 1, 1, 3, 8, 30, 145, 1028, 12320, 274806, 12007355, 1019030239, 165091859656, 50502058492266, 29054157815353374, 31426486309136279775, 64001015806929213894372, 245935864212056913811759534, 1787577725208700551275529005084

Offset: 0

Author

R. J. Mathar and N. J. A. Sloane, Jul 18 2016

Keywords

• nonn

Examples

			a(6) = A216785(6)+2 =30 where the two additional graphs have two equal components (of which there are A001349(3)=2 choices).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..75
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 81.

Crossrefs

Cf. A001349, A216785 (non-isomorphic components), A275165, A275166, column 2 of A201922.

Programs

Mathematica

terms = 20;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
A[x_] = Join[{1}, EULERi[Array[a88, terms]]].x^Range[0, terms] - 1;
CoefficientList[(A[x]^2 + A[x^2])/2 + O[x]^terms, x] (* Jean-François Alcover, Sep 28 2018, after Andrew Howroyd in A001349 *)

Formula

G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A001349 without the initial constant 1.

a(n) = A201922(n,2). - R. J. Mathar, Jul 20 2016