%I #30 Mar 08 2019 15:16:40
%S 1,2,4,22,208,5560,299600,33562696,7635498336,3518440564544,
%T 3275345183542208,6148914696963883712,23248573454127484129024,
%U 176848577040808821410837120,2704321280486889389864215362560,83076749736557243209409446411255936,5124252113632955685095523500148980125696,634332307869315502692705867068871886072665600
%N 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 Suggested by A192314.
%C 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 Vincenzo Librandi, <a href="/A192332/b192332.txt">Table of n, a(n) for n = 1..80</a>
%H Gus Wiseman, <a href="/A192332/a192332.png">Inequivalent representatives of the a(5) = 22 graphical necklaces</a>.
%H Gus Wiseman, <a href="/A192332/a192332_1.png">Inequivalent representatives of the a(5) = 208 graphical necklaces</a>.
%F 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 From _Gus Wiseman_, Mar 04 2019: (Start)
%e Inequivalent representatives of the a(1) = 1 through a(4) = 22 graphical necklace edge-sets:
%e {} {} {} {}
%e {{12}} {{12}} {{12}}
%e {{12}{13}} {{13}}
%e {{12}{13}{23}} {{12}{13}}
%e {{12}{14}}
%e {{12}{24}}
%e {{12}{34}}
%e {{13}{24}}
%e {{12}{13}{14}}
%e {{12}{13}{23}}
%e {{12}{13}{24}}
%e {{12}{13}{34}}
%e {{12}{14}{23}}
%e {{12}{24}{34}}
%e {{12}{13}{14}{23}}
%e {{12}{13}{14}{24}}
%e {{12}{13}{14}{34}}
%e {{12}{13}{24}{34}}
%e {{12}{14}{23}{34}}
%e {{12}{13}{14}{23}{24}}
%e {{12}{13}{14}{23}{34}}
%e {{12}{13}{14}{23}{24}{34}}
%e (End)
%p with(numtheory);
%p f:=proc(n) local t0, t1, d; t0:=0; t1:=divisors(n);
%p for d in t1 do
%p if d mod 2 = 0 then t0:=t0+phi(d)*2^(n^2/(2*d))
%p else t0:=t0+phi(d)*2^(n*(n-1)/(2*d)); fi; od; t0/n; end;
%p [seq(f(n), n=1..30)];
%t Table[ 1/n* Plus @@ Map[Function[d, EulerPhi[d]*2^((n*(n - Mod[d, 2])/2)/d)], Divisors[n]], {n, 1, 20}] (* _Olivier GĂ©rard_, Aug 27 2011 *)
%t rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t 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 (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 Cf. A192314, A191563 (orbits under dihedral group).
%Y Cf. A000031, A000939 (cycle necklaces), A008965, A059966, A060223, A061417, A086675 (digraph version), A184271, A275527, A323858, A324461, A324463, A324464.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jun 28 2011