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

0,5

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.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A059171 A261766 A078619 * A066304 A298456 A145776

Adjacent sequences: A274931 A274932 A274933 * A274935 A274936 A274937

Keyword

nonn

Author

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

Status

approved