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A306715 Number of graphical necklaces with n vertices and distinct rotations.

%I A306715 #9 Aug 15 2019 21:33:03
%S A306715 1,0,2,12,204,5372,299592,33546240,7635496960,3518433853392,
%T A306715 3275345183542176,6148914685509544960,23248573454127484128960,
%U A306715 176848577040728399988915648,2704321280486889389857342715776,83076749736557240903566436660674560
%N A306715 Number of graphical necklaces with n vertices and distinct rotations.
%C A306715 A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. A graphical necklace is a simple graph that is minimal among all n rotations of the vertices.
%H A306715 Andrew Howroyd, <a href="/A306715/b306715.txt">Table of n, a(n) for n = 1..50</a>
%H A306715 Gus Wiseman, <a href="/A306715/a306715.png">Inequivalent representatives of the a(4) = 12 graphical necklaces with distinct rotations</a>.
%F A306715 a(n > 0) = A324461(n)/n.
%F A306715 a(n) = (1/n)*Sum_{d|n} mu(d)*2^(n*(n/d-1)/2 + n*floor(d/2)/d) for n > 0. - _Andrew Howroyd_, Aug 15 2019
%t A306715 rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t A306715 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],With[{rots=Table[Nest[rotgra[#,n]&,#,j],{j,n}]},UnsameQ@@rots&&#==First[Sort[rots]]]&]],{n,5}]
%o A306715 (PARI) a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d))/n)} \\ _Andrew Howroyd_, Aug 15 2019
%Y A306715 Cf. A000088, A001037, A006125, A059966, A060223, A086675, A192332 (graphical necklaces), A306669, A323861, A323865, A323866, A323871, A324461 (distinct rotations), A324513.
%K A306715 nonn
%O A306715 1,3
%A A306715 _Gus Wiseman_, Mar 05 2019
%E A306715 Terms a(7) and beyond from _Andrew Howroyd_, Aug 15 2019