%I #44 Apr 06 2022 05:24:37
%S 1,1,2,3,8,19,57,186,740,3389,18502,120221,932260,8596844,93762704,
%T 1201732437,17992683043,313098431306,6305419392541
%N Number of connected squarefree graphs on n nodes.
%C From _R. J. Mathar_, Apr 04 2022 (Start)
%C The sequence contains the row sums of the number of connected squarefree graphs on V vertices with E edges, the triangle with V>=0, E>=0:
%C 1 ;
%C 1 ;
%C 0 1;
%C 0 0 1 1;
%C 0 0 0 2 1;
%C 0 0 0 0 3 4 1;
%C 0 0 0 0 0 6 9 4;
%C 0 0 0 0 0 0 11 24 17 5;
%C 0 0 0 0 0 0 0 23 61 66 31 5;
%C 0 0 0 0 0 0 0 0 47 169 248 192 74 10;
%C (End)
%H Felix Arends, Joel Ouaknine, and Charles W. Wampler, <a href="https://arxiv.org/abs/1111.3301">On Searching for Small Kochen-Specker Vector Systems</a> (extended version), arXiv:1111.3301 [quant-ph], 2011.
%H CombOS - Combinatorial Object Server, <a href="http://combos.org/nauty">Generate graphs</a>
%H R. J. Mathar, <a href="/A077269/a077269.pdf">Illustrations</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Square-FreeGraph.html">Square-Free Graph</a>
%H <a href="/index/Sq#square_free">Index entries for sequences of squarefree graphs</a>
%F Inverse Euler transform of A006786. - _Andrew Howroyd_, Nov 03 2017
%t A006786 = {1, 2, 4, 8, 18, 44, 117, 351, 1230, 5069, 25181, 152045, 1116403, 9899865, 104980369, 1318017549, 19427531763, 333964672216, 6660282066936};
%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[A006786] (* _Jean-François Alcover_, Aug 18 2018, after _Andrew Howroyd_ *)
%Y Cf. A006786, A243243 (complement).
%K nonn,more
%O 1,3
%A _Eric W. Weisstein_, Nov 01 2002
%E More terms from _Jim Nastos_, Aug 27 2004
%E 4 more terms from _Vladeta Jovovic_, May 17 2008
%E a(18)-a(19) using _Brendan McKay_'s extension to A006786 by _Alois P. Heinz_, Mar 11 2018