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%I #12 Aug 19 2019 16:58:44
%S 1,0,0,3,28,765,26958,1887277,252458904,66376420155,34508978662350,
%T 35645504882731557,73356937843604425644,301275024444053951967585,
%U 2471655539736990372520379226,40527712706903544100966076156895,1328579255614092949957261201822704816
%N Number of simple graphs covering n vertices with distinct rotations.
%C 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. It is covering if there are no isolated vertices. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).
%H Andrew Howroyd, <a href="/A324462/b324462.txt">Table of n, a(n) for n = 0..50</a>
%H Gus Wiseman, <a href="/A324462/a324462.png">The a(4) = 28 graph covers with distinct rotations</a>.
%H Gus Wiseman, <a href="/A324462/a324462_1.png">The a(4) = 28 graph covers with distinct rotations, radial embedding</a>.
%F a(n) = Sum{d|n} mu(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ) for n > 0. - _Andrew Howroyd_, Aug 19 2019
%t rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]]&]],{n,0,5}]
%o (PARI) a(n)={if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d,k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2)))))} \\ _Andrew Howroyd_, Aug 19 2019
%Y Cf. A000088, A000740, A002494, A006125, A006129, A027375, A192332, A323863, A323864, A323867, A323869, A324461 (non-covering case), A324463, A324464.
%K nonn
%O 0,4
%A _Gus Wiseman_, Feb 28 2019
%E Terms a(7) and beyond from _Andrew Howroyd_, Aug 19 2019