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Number of necklaces with n white beads and 9*n black beads.
2

%I #16 Apr 30 2019 08:22:56

%S 1,1,10,136,2290,42376,834472,17125354,362345362,7847250409,

%T 173103115760,3875837737520,87857163416200,2012268157890524,

%U 46497242510288804,1082614775186919136,25374686712458407442,598217593341171422090,14176230568954212214234

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

%H Alois P. Heinz, <a href="/A261503/b261503.txt">Table of n, a(n) for n = 0..710</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/(10*n) * Sum_{d|n} C(10*n/d,n/d) * A000010(d) for n>0, a(0) = 1.

%F a(n) ~ 10^(10*n-1/2) / (sqrt(2*Pi) * 9^(9*n+1/2) * n^(3/2)). - _Vaclav Kotesovec_, Aug 22 2015

%p with(numtheory):

%p a:= n-> `if`(n=0, 1, add(binomial(10*n/d, n/d)

%p *phi(d), d=divisors(n))/(10*n)):

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

%Y Column k=9 of A261494.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 21 2015