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A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention.

%I A192332 #30 Mar 08 2019 15:16:40
%S A192332 1,2,4,22,208,5560,299600,33562696,7635498336,3518440564544,
%T A192332 3275345183542208,6148914696963883712,23248573454127484129024,
%U A192332 176848577040808821410837120,2704321280486889389864215362560,83076749736557243209409446411255936,5124252113632955685095523500148980125696,634332307869315502692705867068871886072665600
%N A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention.
%C A192332 Suggested by A192314.
%C A192332 Also the number of graphical necklaces with n vertices. We define a graphical necklace to be a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. These are a kind of partially labeled graphs. - _Gus Wiseman_, Mar 04 2019
%H A192332 Vincenzo Librandi, <a href="/A192332/b192332.txt">Table of n, a(n) for n = 1..80</a>
%H A192332 Gus Wiseman, <a href="/A192332/a192332.png">Inequivalent representatives of the a(5) = 22 graphical necklaces</a>.
%H A192332 Gus Wiseman, <a href="/A192332/a192332_1.png">Inequivalent representatives of the a(5) = 208 graphical necklaces</a>.
%F A192332 a(n) = (1/n)*(Sum_{d|n, d odd} phi(d)*2^(n*(n-1)/(2*d)) + Sum_{d|n, d even} phi(d)*2^(n^2/(2*d))).
%e A192332 From _Gus Wiseman_, Mar 04 2019: (Start)
%e A192332 Inequivalent representatives of the a(1) = 1 through a(4) = 22 graphical necklace edge-sets:
%e A192332   {}  {}      {}              {}
%e A192332       {{12}}  {{12}}          {{12}}
%e A192332               {{12}{13}}      {{13}}
%e A192332               {{12}{13}{23}}  {{12}{13}}
%e A192332                               {{12}{14}}
%e A192332                               {{12}{24}}
%e A192332                               {{12}{34}}
%e A192332                               {{13}{24}}
%e A192332                               {{12}{13}{14}}
%e A192332                               {{12}{13}{23}}
%e A192332                               {{12}{13}{24}}
%e A192332                               {{12}{13}{34}}
%e A192332                               {{12}{14}{23}}
%e A192332                               {{12}{24}{34}}
%e A192332                               {{12}{13}{14}{23}}
%e A192332                               {{12}{13}{14}{24}}
%e A192332                               {{12}{13}{14}{34}}
%e A192332                               {{12}{13}{24}{34}}
%e A192332                               {{12}{14}{23}{34}}
%e A192332                               {{12}{13}{14}{23}{24}}
%e A192332                               {{12}{13}{14}{23}{34}}
%e A192332                               {{12}{13}{14}{23}{24}{34}}
%e A192332 (End)
%p A192332 with(numtheory);
%p A192332 f:=proc(n) local t0, t1, d; t0:=0; t1:=divisors(n);
%p A192332 for d in t1 do
%p A192332 if d mod 2 = 0 then t0:=t0+phi(d)*2^(n^2/(2*d))
%p A192332 else t0:=t0+phi(d)*2^(n*(n-1)/(2*d)); fi; od; t0/n; end;
%p A192332 [seq(f(n), n=1..30)];
%t A192332 Table[ 1/n* Plus @@ Map[Function[d, EulerPhi[d]*2^((n*(n - Mod[d, 2])/2)/d)], Divisors[n]], {n, 1, 20}]  (* _Olivier Gérard_, Aug 27 2011 *)
%t A192332 rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t A192332 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,0,5}] (* _Gus Wiseman_, Mar 04 2019 *)
%o A192332 (PARI) a(n) = sumdiv(n, d, if (d%2, eulerphi(d)*2^(n*(n-1)/(2*d)), eulerphi(d)*2^(n^2/(2*d))))/n; \\ _Michel Marcus_, Mar 08 2019
%Y A192332 Cf. A192314, A191563 (orbits under dihedral group).
%Y A192332 Cf. A000031, A000939 (cycle necklaces), A008965, A059966, A060223, A061417, A086675 (digraph version), A184271, A275527, A323858, A324461, A324463, A324464.
%K A192332 nonn
%O A192332 1,2
%A A192332 _N. J. A. Sloane_, Jun 28 2011