A214311 - OEIS

%I #20 Sep 26 2017 14:47:40

%S 12,30,150,633,3260,16212,66810,298495,1410402,6403842,31103899,

%T 135342046,633228696,2936824916,13676037486,65355191817,298065986582,

%U 1398226666434,6585151203697,30958838054304,148994847644780

%N a(n) is the number of representative five-color bracelets (necklaces with turning over allowed) with n beads, for n >= 5.

%C This is the fifth column (m=5) of triangle A213940.

%C The relevant p(n,5)= A008284(n,5) representative color multinomials have exponents (signatures) from the five-part partitions of n, written with nonincreasing parts. E.g., n=7: [3,1,1,1,1] and [2,2,1,1,1] (p(7,5)=2). The corresponding representative bracelets have the five-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5] and c[1]^2*c[2]^2*c[3]*c[4]*c[5].

%C Number of n-length bracelets w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226884). The number of 5 color bracelets up to permutations of colors is given by A056360. - _Andrew Howroyd_, Sep 26 2017

%H Andrew Howroyd, <a href="/A214311/b214311.txt">Table of n, a(n) for n = 5..200</a>

%F a(n) = A213940(n,5), n >= 5.

%F a(n) = sum(A213939(n,k),k= b(n,5)..b(n,6)-1), n>=6, with b(n,m) = A214314(n,m) the position where the first m part partition of n appears in the Abramowitz-Stegun ordering of partitions (see A036036 for the reference and a historical comment). a(5) = A213939(5,b(5,5)) =  A213939(5,7) = 12.

%e a(5) = A213940(5,5) = A213939(5,7) = 12 from the representative bracelets (with colors j for c[j], j=1,...,5) 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235 and 14325, all taken cyclically.

%Y Cf. A213939, A213940, A214309 (m=4 case), A214313 (m=5, all bracelets).

%Y Cf. A056360, A226884.

%K nonn

%O 5,1

%A _Wolfdieter Lang_, Aug 08 2012