A261503 Number of necklaces with n white beads and 9*n black beads.

1, 1, 10, 136, 2290, 42376, 834472, 17125354, 362345362, 7847250409, 173103115760, 3875837737520, 87857163416200, 2012268157890524, 46497242510288804, 1082614775186919136, 25374686712458407442, 598217593341171422090, 14176230568954212214234

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..710
• 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=9 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(10*n/d, n/d)
             *phi(d), d=divisors(n))/(10*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(10*n) * Sum_{d|n} C(10*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 10^(10*n-1/2) / (sqrt(2*Pi) * 9^(9*n+1/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015