A214311 a(n) is the number of representative five-color bracelets (necklaces with turning over allowed) with n beads, for n >= 5.

12, 30, 150, 633, 3260, 16212, 66810, 298495, 1410402, 6403842, 31103899, 135342046, 633228696, 2936824916, 13676037486, 65355191817, 298065986582, 1398226666434, 6585151203697, 30958838054304, 148994847644780

Offset

5,1

Comments

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

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].

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

Links

Andrew Howroyd, Table of n, a(n) for n = 5..200

Formula

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

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.

Example

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.

Crossrefs

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

Cf. A056360, A226884.

Sequence in context: A117313 A080563 A221520 * A005147 A320122 A007308

Adjacent sequences: A214308 A214309 A214310 * A214312 A214313 A214314

Keyword

nonn

Author

Wolfdieter Lang, Aug 08 2012

Status

approved