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%I #31 Feb 22 2019 10:16:13
%S 1,1,1,3,6,27,108,788,9035,196928,7797877,533938066
%N Number of simple connected well-covered graphs on n nodes.
%C A graph is well-covered if its maximal independent vertex sets are of equal size.
%H Jonathan Baker, K. N. V. Meulen, A. Van Tuyl, <a href="https://arxiv.org/abs/1606.04447">Shedding vertices of vertex decomposable graphs</a>, arXiv preprint arXiv:1606.04447 [math.CO], 2016.
%H F. Hüffner, <a href="https://github.com/falk-hueffner/tinygraph">tinygraph</a>, software for generating integer sequences based on graph properties, version a1db88e
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Well-CoveredGraph.html">Well-Covered Graph</a>
%F a(n) = A222626(n) - A287025(n).
%F Inverse Euler transform of A222626. - _Andrew Howroyd_, Nov 03 2017
%t A222626 = {1, 2, 3, 7, 14, 46, 164, 996, 10195, 208168, 8016530, 542165050};
%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 EULERi[A222626] (* _Jean-François Alcover_, Feb 22 2019, after _Andrew Howroyd_ *)
%Y Cf. A222626 (number of simple not-necessarily connected well-covered graphs).
%Y Cf. A287025 (number of simple disconnected well-covered graphs).
%K nonn,more
%O 1,4
%A _Eric W. Weisstein_, Feb 26 2013
%E a(10) from _Eric W. Weisstein_, Aug 29 2013
%E a(11)-a(12) added using tinygraph by _Falk Hüffner_, Aug 15 2017
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