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A056291
Number of primitive (period n) n-bead necklaces with exactly six different colored beads.
4
0, 0, 0, 0, 0, 120, 2160, 23940, 211680, 1643544, 11748240, 79419060, 516257280, 3262440960, 20193277104, 123071683140, 741419995680, 4427489935680, 26264144909520, 155018839412052, 911509010152560, 5344538372696880, 31272099902089200, 182707081042818360
OFFSET
1,6
COMMENTS
Turning over the necklace is not allowed.
REFERENCES
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
LINKS
FORMULA
Sum mu(d)*A056286(n/d) where d|n.
MAPLE
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 6-j)*binomial(6, j)*(-1)^j, j=0..6):
seq(a(n), n=1..30); # Alois P. Heinz, Jan 25 2015
MATHEMATICA
b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
a[n_] := Sum[b[n, 6 - j]*Binomial[6, j]*(-1)^j, {j, 0, 6}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)
CROSSREFS
Cf. A032164.
Column k=6 of A254040.
Sequence in context: A293972 A144858 A084030 * A056286 A166779 A038745
KEYWORD
nonn
STATUS
approved