A261499 Number of necklaces with n white beads and 5*n black beads.

1, 1, 6, 46, 446, 4751, 54132, 642342, 7861662, 98480332, 1256569506, 16273981758, 213378976004, 2826867619109, 37782553160820, 508840821830546, 6898459216311582, 94070535317459018, 1289430373206452136, 17755914760643605782, 245518560760433583946

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..850
• 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=5 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(6*n/d, n/d)
             *phi(d), d=divisors(n))/(6*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(6*n) * Sum_{d|n} C(6*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 6^(6*n) / (2 * sqrt(3*Pi) * 5^(5*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015