A261502 Number of necklaces with n white beads and 8*n black beads.

1, 1, 9, 109, 1641, 27151, 478341, 8782075, 166237161, 3220837534, 63562741159, 1273237637706, 25820645555109, 529080420540114, 10937268142896643, 227824992158991334, 4777204094770874857, 100757627271124231383, 2136117417348870713646, 45496022230420668679932

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..730
• 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=8 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(9*n/d, n/d)
             *phi(d), d=divisors(n))/(9*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(9*n) * Sum_{d|n} C(9*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 9^(9*n-1/2) / (sqrt(2*Pi) * 8^(8*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015