A261501 Number of necklaces with n white beads and 7*n black beads.

1, 1, 8, 85, 1128, 16451, 255704, 4141383, 69159400, 1182125128, 20581159608, 363704640476, 6506965279992, 117626432708864, 2145180358634664, 39421026305282660, 729242353169440744, 13568988503585900648, 253785064586356459616, 4768543107831461199897

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..760
• 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=7 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(8*n/d, n/d)
             *phi(d), d=divisors(n))/(8*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(8*n) * Sum_{d|n} C(8*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 2^(24*n-2) / (sqrt(Pi) * 7^(7*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015