A073020 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.

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, 1, 8, 21, 72, 147, 280, 375, 440, 1, 9, 27, 104, 252, 561, 912, 1282, 1387, 1, 10, 33, 145, 406, 1032, 1980, 3260, 4262, 4752, 1, 11, 40, 195, 621, 1782, 3936, 7440, 11410

Offset: 1

Author

Wouter Meeussen, Aug 03 2002

Keywords

• nonn
• tabl

Comments

Left half of even rows of table A052307 with left column deleted.

Examples

			1; 1,2; 1,3,3; 1,4,5,8; 1,5,8,16,16; ...
		

Links

• Index entries for sequences related to bracelets

Crossrefs

Cf. A052307, A047996, A072506, A005648. Cf. A078925 for odd number of beads. Last term in each row gives A005648.

Programs

Mathematica

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}]
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}]
Table[ f[k, 2n], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v, Mar 29 2006 *)

Formula

(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) ).