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A203398 T(n,k), a triangular array read by rows, is the number of classes of equivalent 2-color n-bead necklaces (turning over is not allowed) that have k necklaces.

%I A203398 #21 Jun 07 2019 19:39:16
%S A203398 2,2,1,2,0,2,2,1,0,3,2,0,0,0,6,2,1,2,0,0,9,2,0,0,0,0,0,18,2,1,0,3,0,0,
%T A203398 0,30,2,0,2,0,0,0,0,0,56,2,1,0,0,6,0,0,0,0,99,2,0,0,0,0,0,0,0,0,0,186,
%U A203398 2,1,2,3,0,9,0,0,0,0,0,335
%N A203398 T(n,k), a triangular array read by rows, is the number of classes of equivalent 2-color n-bead necklaces (turning over is not allowed) that have k necklaces.
%C A203398 Equivalently, the cyclic group of order n acts on the set of length n binary sequences.  T(n,k) is the number of orbits that have k elements.
%H A203398 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A203398 Frank Ruskey, <a href="https://web.archive.org/web/20171022155546/http://www.1stworks.com/ref/RuskeyCombGen.pdf">Combinatorial Generation Algorithm Algorithm 4.24, p. 95</a>.
%e A203398   2
%e A203398   2  1
%e A203398   2  0  2
%e A203398   2  1  0  3
%e A203398   2  0  0  0  6
%e A203398   2  1  2  0  0  9
%e A203398   2  0  0  0  0  0  18
%e A203398   2  1  0  3  0  0  0  30
%e A203398   2  0  2  0  0  0  0  0  56
%e A203398   2  1  0  0  6  0  0  0  0  99
%e A203398   2  0  0  0  0  0  0  0  0  0  186
%e A203398   2  1  2  3  0  9  0  0  0  0  0   335
%t A203398 Needs["Combinatorica`"];
%t A203398 f[list_] := Sort[NestList[RotateLeft, list, Length[list]-1]]; Flatten[Table[Distribution[Map[Length, Map[Union, Union[Map[f, Strings[{0, 1}, n]]]]], Range[n]], {n, 1, 12}]]
%Y A203398 A000031 (row sums), T(n,n) = A001037, T(n,n) = A064535 when n is prime, T(n,k) = A001037(k) when k divides n.
%Y A203398 Cf. A203399.
%K A203398 nonn,tabl
%O A203398 1,1
%A A203398 _Geoffrey Critzer_, Jan 01 2012