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

0,3

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

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

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);

Crossrefs

Column k=7 of A261494.

Sequence in context: A300675 A241323 A349334 * A371897 A180582 A230621

Adjacent sequences: A261498 A261499 A261500 * A261502 A261503 A261504

Keyword

nonn

Author

Alois P. Heinz, Aug 21 2015

Status

approved