

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.


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
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OFFSET

2,2


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 kth partition of n in AbramowitzStegun (ASt) 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.


LINKS

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


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


EXAMPLE

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].


CROSSREFS

Cf. A213941, A214306, A213942 (m=2, representative bracelets), A214310 (m=3).
Sequence in context: A258345 A258351 A130669 * A237350 A265393 A292908
Adjacent sequences: A214305 A214306 A214307 * A214309 A214310 A214311


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Jul 31 2012


EXTENSIONS

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


STATUS

approved



