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A324462 Number of simple graphs covering n vertices with distinct rotations.

%I A324462 #12 Aug 19 2019 16:58:44
%S A324462 1,0,0,3,28,765,26958,1887277,252458904,66376420155,34508978662350,
%T A324462 35645504882731557,73356937843604425644,301275024444053951967585,
%U A324462 2471655539736990372520379226,40527712706903544100966076156895,1328579255614092949957261201822704816
%N A324462 Number of simple graphs covering n vertices with distinct rotations.
%C A324462 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 A324462 Andrew Howroyd, <a href="/A324462/b324462.txt">Table of n, a(n) for n = 0..50</a>
%H A324462 Gus Wiseman, <a href="/A324462/a324462.png">The a(4) = 28 graph covers with distinct rotations</a>.
%H A324462 Gus Wiseman, <a href="/A324462/a324462_1.png">The a(4) = 28 graph covers with distinct rotations, radial embedding</a>.
%F A324462 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 A324462 rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t A324462 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]]&]],{n,0,5}]
%o A324462 (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 A324462 Cf. A000088, A000740, A002494, A006125, A006129, A027375, A192332, A323863, A323864, A323867, A323869, A324461 (non-covering case), A324463, A324464.
%K A324462 nonn
%O A324462 0,4
%A A324462 _Gus Wiseman_, Feb 28 2019
%E A324462 Terms a(7) and beyond from _Andrew Howroyd_, Aug 19 2019