A261500 Number of necklaces with n white beads and 6*n black beads.

1, 1, 7, 64, 735, 9276, 124936, 1753074, 25366335, 375677659, 5667212132, 86775157140, 1345153548264, 21069043965984, 332927800269694, 5301031234085664, 84967018635587775, 1369846562874360887, 22199151536133457885, 361411377745122110422, 5908312923795257331460

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Eric Weisstein's World of Mathematics, Necklace
• Wikipedia, Necklace (combinatorics)
• Index entries for sequences related to necklaces

Crossrefs

Column k=6 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(7*n/d, n/d)
             *phi(d), d=divisors(n))/(7*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(7*n) * Sum_{d|n} C(7*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 7^(7*n-1/2) / (2 * sqrt(3*Pi) * 6^(6*n) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015