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A292223 a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6. 0
60, 180, 1050, 5040, 29244, 161340, 1046250, 4825800, 27790266, 145126548, 843333015, 4466836920, 26967624184, 137243187108, 789854179074, 4306147750200, 24711052977222, 134216193832908, 797987818325009, 4240082199867228 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,1

COMMENTS

This is the sixth column (m = 6) of triangle A213940.

The relevant p(n,6)= A008284(n, 6) representative color multinomials have exponents (signatures) from the six-part partitions of n, written with nonincreasing parts. E.g., n = 8: [3,1,1,1,1,1] and [2,2,1,1,1,1] (p(8,6)=2). The corresponding representative bracelets have the six-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5]*c[6] and c[1]^2*c[2]^2*c[3]*c[4]*c[5]*c[6].

See A056361 for the numbers if also color permutations for D_6 inequivalent bracelets are allowed. (Andrew Howroyd induced me to look at these bracelets.)

LINKS

Table of n, a(n) for n=6..25.

FORMULA

a(n) = A213940(n, 6), n >= 6.

a(n) = Sum_{k=b(n, 6)..b(n, 7)-1} A213939(n, k), for n >= 7, 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), and a(6) = A213939(6, b(6,6)) = A213939(6, 11) = 60.

EXAMPLE

a(6) = A213940(6,6) = A213939(6, 11) = 60 from the representative bracelets (with colors j for c(j), j=1..6) permutations of (1, 2, 3, 4, 5, 6) modulo D_6 (dihedral group) symmetry, i.e., modulo cyclic or anti-cyclic operations. E.g., (1, 2, 3, 4, 6, 5) == (2, 3, 4, 6, 5, 1) == (6, 4, 3, 2, 1, 5) == ..., but (1, 2, 3, 4, 6, 5) is not equivalent to (1, 2, 3, 4, 5, 6). If color permutation is also allowed, then there is only one possibility (see A056361(6) = 1).

CROSSREFS

Cf. A008284, A036036, A056361, A213940, A214311, A214314.

Sequence in context: A291549 A259946 A249911 * A112827 A181333 A082529

Adjacent sequences:  A292220 A292221 A292222 * A292224 A292225 A292226

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Sep 30 2017

STATUS

approved

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Last modified September 20 12:42 EDT 2018. Contains 315239 sequences. (Running on oeis4.)