

A292223


a(n) is the number of representative sixcolor 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
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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 sixpart 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 sixcolor 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 mpart partition of n appears in the AbramowitzStegun 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 anticyclic 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



