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A274934 Number of unlabeled graphs with n nodes that have two components, neither of which is the empty graph. 10
0, 0, 1, 1, 3, 8, 30, 145, 1028, 12320, 274806, 12007355, 1019030239, 165091859656, 50502058492266, 29054157815353374, 31426486309136279775, 64001015806929213894372, 245935864212056913811759534, 1787577725208700551275529005084 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
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
KEYWORD
nonn
AUTHOR
R. J. Mathar and N. J. A. Sloane, Jul 18 2016
STATUS
approved

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Last modified March 2 14:01 EST 2024. Contains 370485 sequences. (Running on oeis4.)