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%I #33 May 27 2021 17:08:54
%S 1,2,3,7,12,37,93,354,1350,6351,31950,179307,1071265,6845581,46162583,
%T 327731950,2437753740,18948599220,153498350745,1293123243928,
%U 11306475314467,102425554299516,959826755336242,9290811905391501
%N Number of set partitions up to rotations and reflections.
%C Combines the symmetry operations of A080107 and A084423.
%C Equivalently, number of n-bead bracelets using any number of unlabeled (interchangable) colors. - _Andrew Howroyd_, Sep 25 2017
%H Michael De Vlieger, <a href="/A084708/b084708.txt">Table of n, a(n) for n = 1..577</a>
%H Colin Adams, Chaim Even-Zohar, Jonah Greenberg, Reuben Kaufman, David Lee, Darin Li, Dustin Ping, Theodore Sandstrom, and Xiwen Wang, <a href="https://arxiv.org/abs/2103.08314">Virtual Multicrossings and Petal Diagrams for Virtual Knots and Links</a>, arXiv:2103.08314 [math.GT], 2021.
%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#rotref">Partition related number triangles</a>
%H N. J. A. Sloane, <a href="/A000013/a000013.txt">Generating functions</a> [From _Wouter Meeussen_, Dec 06 2008]
%F a(n) = (A080107(n)+A084423(n))/2. - _Wouter Meeussen_ and _Vladeta Jovovic_, Nov 28 2008
%e SetPartitions[6] is the first to decompose differently from A084423: 4 cycles of length 1, 2 of 2, 9 of 3, 16 of 6, 6 of 12.
%e a(7) = 1 + A056357(7) + A056358(7) + A056359(7) + A056360(7) + A056361(7) + 1 = 1 + 8 + 31 + 33 + 16 + 3 + 1 = 93.
%t <<DiscreteMath`NewCombinatorica`; (* see A080107 *); Table[{Length[ # ], First[ # ]}&/@ Split[Sort[Length/@Split[Sort[First[Sort[Flatten[ {#, Map[Sort, (#/. i_Integer:>w+1-i), 2]}& @(NestList[Sort[Sort/@(#/. i_Integer :> Mod[i+1, w, 1])]&, #, w]), 1]]]&/@SetPartitions[w]]]]], {w, 1, 10}]
%t u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; a[n_]:=1/n*Plus@@(EulerPhi[ # ]u[Quotient[n,# ],# ]&/@Divisors[n]); Table[a[n]/2+If[EvenQ[n],u[n/2,2],Sum[Binomial[n/2-1/2,k] u[k,2], {k,0,n/2-1/2}]]/2,{n,40}] (* _Wouter Meeussen_, Dec 06 2008 *)
%Y Cf. A080107, A084423, A080510, A002872, A002874, A141003, A036075, A141004, A036077, A152176.
%K nonn
%O 1,2
%A _Wouter Meeussen_, Jul 02 2003
%E a(12) from _Vladeta Jovovic_, Jul 15 2007
%E More terms from _Vladeta Jovovic_'s formula given in Mathematica line. - _Wouter Meeussen_, Dec 06 2008
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