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%I #30 Sep 28 2018 19:15:46
%S 0,0,1,1,3,8,30,145,1028,12320,274806,12007355,1019030239,
%T 165091859656,50502058492266,29054157815353374,31426486309136279775,
%U 64001015806929213894372,245935864212056913811759534,1787577725208700551275529005084
%N Number of unlabeled graphs with n nodes that have two components, neither of which is the empty graph.
%H Alois P. Heinz, <a href="/A274934/b274934.txt">Table of n, a(n) for n = 0..75</a>
%H R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO] (2017) Table 81.
%F G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A001349 without the initial constant 1.
%F a(n) = A201922(n,2). - _R. J. Mathar_, Jul 20 2016
%e a(6) = A216785(6)+2 =30 where the two additional graphs have two equal components (of which there are A001349(3)=2 choices).
%t terms = 20;
%t mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t 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]];
%t 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];
%t edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
%t a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
%t A[x_] = Join[{1}, EULERi[Array[a88, terms]]].x^Range[0, terms] - 1;
%t CoefficientList[(A[x]^2 + A[x^2])/2 + O[x]^terms, x] (* _Jean-François Alcover_, Sep 28 2018, after _Andrew Howroyd_ in A001349 *)
%Y Cf. A001349, A216785 (non-isomorphic components), A275165, A275166, column 2 of A201922.
%K nonn
%O 0,5
%A _R. J. Mathar_ and _N. J. A. Sloane_, Jul 18 2016
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