A214308 a(n) is the number of all two colored bracelets (necklaces with turning over allowed) with n beads with the two colors from a repertoire of n distinct colors, for n>=2.

1, 6, 24, 60, 165, 336, 784, 1584, 3420, 6820, 14652, 29484, 62335, 128310, 269760, 558960, 1175499, 2446668, 5131900, 10702020, 22385517, 46655224, 97344096, 202555800, 421478200, 875297124, 1816696728, 3764747868, 7795573230, 16121364000, 33310887808

Offset: 2

Author

Wolfdieter Lang, Jul 31 2012

Keywords

• nonn

Comments

This is the second column (m=2) of triangle A214306.

Each 2 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2]], with p[1] >= p[2] >= 1, there are A213941(n,k)= A035206(n,k)*A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,2)= A008284(n,2) = floor(n/2) partitions of n with 2 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].

Compare this sequence with A000029 where also single colored bracelets are included, and the color repertoire is only [c[1], c[2]] for all n.

Examples

			a(5) = A213941(5,2) + A213941(5,3) = 20 + 40 = 60 from the bracelet (with colors j for c[j], j=1,2,..,5) cyclic(11112) which represents a class of order A035206(5,2) = 20 (if all 5 colors are used), cyclic(11122) and cyclic(11212) each representing also a color class of 20 members each, summing to 60 bracelets  with five beads and five colors available for the two color signatures [4,1] and [3,2].
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..100

Crossrefs

Cf. A213941, A214306, A213942 (m=2, representative bracelets), A214310 (m=3).

Formula

a(n) = A214306(n,2), n >= 2.

a(n) = sum(A213941(n,k),k=2..A008284(n,2)+1), n>=2, with A008284(n,2) = floor(n/2).

a(n) = binomial(n,2) * A056342(n). - Andrew Howroyd, Mar 25 2017

Extensions

a(25)-a(32) from Andrew Howroyd, Mar 25 2017