A077269 - OEIS

%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