cp's OEIS Frontend

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.

%I A214308 #16 Mar 26 2017 07:24:12
%S A214308 1,6,24,60,165,336,784,1584,3420,6820,14652,29484,62335,128310,269760,
%T A214308 558960,1175499,2446668,5131900,10702020,22385517,46655224,97344096,
%U A214308 202555800,421478200,875297124,1816696728,3764747868,7795573230,16121364000,33310887808
%N 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.
%C A214308 This is the second column (m=2) of triangle A214306.
%C A214308 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]].
%C A214308 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.
%H A214308 Andrew Howroyd, <a href="/A214308/b214308.txt">Table of n, a(n) for n = 2..100</a>
%F A214308 a(n) = A214306(n,2), n >= 2.
%F A214308 a(n) = sum(A213941(n,k),k=2..A008284(n,2)+1), n>=2, with A008284(n,2) = floor(n/2).
%F A214308 a(n) = binomial(n,2) * A056342(n). - _Andrew Howroyd_, Mar 25 2017
%e A214308 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].
%Y A214308 Cf. A213941, A214306, A213942 (m=2, representative bracelets), A214310 (m=3).
%K A214308 nonn
%O A214308 2,2
%A A214308 _Wolfdieter Lang_, Jul 31 2012
%E A214308 a(25)-a(32) from _Andrew Howroyd_, Mar 25 2017