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Number of unlabeled, connected graphs on n vertices with at least one subgraph isomorphic to a C_4, where C_4 is the cycle graph on four vertices.
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%I #33 Feb 15 2019 16:25:37

%S 0,0,0,3,13,93,796,10931,260340,11713182,1006682063,164059710255,

%T 50335906936959,29003487454251217,31397381142667479256,

%U 63969560113223974443840,245871831682084008526845525,1787331725248899088577102145274,24636021429399867655316345340289103

%N Number of unlabeled, connected graphs on n vertices with at least one subgraph isomorphic to a C_4, where C_4 is the cycle graph on four vertices.

%H Travis Hoppe and Anna Petrone, <a href="https://github.com/thoppe/Encyclopedia-of-Finite-Graphs">Encyclopedia of Finite Graphs</a>

%H T. Hoppe and A. Petrone, <a href="http://arxiv.org/abs/1408.3644">Integer sequence discovery from small graphs</a>, arXiv preprint arXiv:1408.3644 [math.CO], 2014.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Square-FreeGraph.html">Square-Free Graph</a>

%F a(n) = A001349(n) - A077269(n).

%t terms = 19;

%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 A001349 = EULERi[Array[a88, terms]];

%t A006786 = {1, 2, 4, 8, 18, 44, 117, 351, 1230, 5069, 25181, 152045, 1116403, 9899865, 104980369, 1318017549, 19427531763, 333964672216, 6660282066936};

%t A077269 = EULERi[A006786];

%t A001349 - A077269 (* _Jean-François Alcover_, Feb 15 2019, after _Andrew Howroyd_ in A001349 and A077269 *)

%Y Cf. A001349, A077269.

%K nonn

%O 1,4

%A _Travis Hoppe_ and _Anna Petrone_, Jun 01 2014

%E a(11)-a(17) using formula from _Falk Hüffner_, Jan 15 2016

%E a(18)-a(19) from _Jean-François Alcover_, Feb 15 2019 using _Andrew Howroyd_'s code.