A261496 Number of necklaces with n white beads and n^2-n black beads.

1, 1, 2, 10, 116, 2126, 54132, 1753074, 69159400, 3220837534, 173103115760, 10551652603526, 719578430426044, 54297978110913252, 4492502634679508204, 404469190271900056316, 39370123445405248353744, 4120204305690280446004838, 461365717080849798202175772

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..335
• 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

Lower diagonal of A261494.

Cf. A000010.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(n^2/d, n/d)
            *phi(d), d=divisors(n))/n^2):
seq(a(n), n=0..20);

Formula

a(n) = 1/(n^2) * Sum_{d|n} C(n^2/d,n/d) * A000010(d) for n>0, a(0) = 1.

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