%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