A261498 Number of necklaces with n white beads and 4*n black beads.

1, 1, 5, 31, 245, 2126, 19811, 192130, 1922741, 19692535, 205446630, 2175519380, 23322657491, 252631900236, 2760768051914, 30400169157656, 336977765092789, 3757141504436393, 42107201595510563, 474084628585822413, 5359833704140820870, 60823006052351729266

Offset: 0

Author

Alois P. Heinz, Aug 21 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..920
• 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=4 of A261494.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(binomial(5*n/d, n/d)
             *phi(d), d=divisors(n))/(5*n)):
seq(a(n), n=0..25);

Formula

a(n) = 1/(5*n) * Sum_{d|n} C(5*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

a(n) ~ 5^(5*n-1/2) / (sqrt(Pi) * 2^(8*n+3/2) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2015