Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #10 Jul 19 2015 01:11:36
%S 1,1,2,1,3,3,1,4,5,8,1,5,8,16,16,1,6,12,29,38,50,1,7,16,47,79,126,133,
%T 1,8,21,72,147,280,375,440,1,9,27,104,252,561,912,1282,1387,1,10,33,
%U 145,406,1032,1980,3260,4262,4752,1,11,40,195,621,1782,3936,7440,11410
%N Triangle of T(n,m) = number of bracelets (necklaces than can be turned over) with m white beads and (2n-m) black ones, for 1<=m<=n.
%C Left half of even rows of table A052307 with left column deleted.
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F (1/2)*(C(2*(n\2), m\2) +Sum (d|(2n, m) phi(d)C(2n/d, m/d) ) - (-1)^n if(even(n+m), 0, C(n-1, floor(m/2-1/2) ).
%e 1; 1,2; 1,3,3; 1,4,5,8; 1,5,8,16,16; ...
%t Table[Length[ Union[Last[Sort[Flatten[Table[{RotateLeft[ #, i], Reverse[RotateLeft[ #, i]]}, {i, 2k}], 1]]]& /@ Permutations[IntegerDigits[2^(2k-j) (2^j-1), 2]]] ], {k, 9}, {j, k}]
%t Table[( -(-1)^n If[EvenQ[m+n], 0, Binomial[n-1, Floor[(m-2)/2]] ]/2 + Fold[ #1+EulerPhi[ #2]Binomial[2n/#2, m/#2]/(2n)&, Binomial[2Floor[n/2], Floor[m/2]], Intersection[Divisors[2n], Divisors[m]]]/2), {n, 9}, {m, n}]
%t Table[ f[k, 2n], {n, 11}, {k, n}] // Flatten (* _Robert G. Wilson v_, Mar 29 2006 *)
%Y Cf. A052307, A047996, A072506, A005648. Cf. A078925 for odd number of beads. Last term in each row gives A005648.
%K nonn,tabl
%O 1,3
%A _Wouter Meeussen_, Aug 03 2002