A261498 - OEIS

%I #16 Apr 30 2019 08:23:29

%S 1,1,5,31,245,2126,19811,192130,1922741,19692535,205446630,2175519380,

%T 23322657491,252631900236,2760768051914,30400169157656,

%U 336977765092789,3757141504436393,42107201595510563,474084628585822413,5359833704140820870,60823006052351729266

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

%H Alois P. Heinz, <a href="/A261498/b261498.txt">Table of n, a(n) for n = 0..920</a>

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Necklace.html">Necklace</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Necklace_(combinatorics)">Necklace (combinatorics)</a>

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

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

%F a(n) ~ 5^(5*n-1/2) / (sqrt(Pi) * 2^(8*n+3/2) * n^(3/2)). - _Vaclav Kotesovec_, Aug 22 2015

%p with(numtheory):

%p a:= n-> `if`(n=0, 1, add(binomial(5*n/d, n/d)

%p              *phi(d), d=divisors(n))/(5*n)):

%p seq(a(n), n=0..25);

%Y Column k=4 of A261494.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 21 2015