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A261504 Number of necklaces with n white beads and 10*n black beads.

%I A261504 #21 Feb 16 2025 08:33:26
%S A261504 1,1,11,166,3091,63251,1376738,31231278,730421043,17485161178,
%T A261504 426342182761,10551652603526,264383070217234,6693356751004275,
%U A261504 170957058318902538,4399826606699783666,113989308154754290611,2970465160237643057033,77808659386282667142686
%N A261504 Number of necklaces with n white beads and 10*n black beads.
%C A261504 In general, for k>=1 is column k of A261494 asymptotic to (k+1)^((k+1)*n-1/2) / (sqrt(2*Pi) * k^(k*n+1/2) * n^(3/2)). - _Vaclav Kotesovec_, Aug 22 2015
%H A261504 Alois P. Heinz, <a href="/A261504/b261504.txt">Table of n, a(n) for n = 0..690</a>
%H A261504 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A261504 F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A261504 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Necklace.html">Necklace</a>
%H A261504 Wikipedia, <a href="https://en.wikipedia.org/wiki/Necklace_(combinatorics)">Necklace (combinatorics)</a>
%H A261504 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F A261504 a(n) = 1/(11*n) * Sum_{d|n} C(11*n/d,n/d) * A000010(d) for n>0, a(0) = 1.
%F A261504 a(n) ~ 11^(11*n-1/2) / (sqrt(2*Pi) * 10^(10*n+1/2) * n^(3/2)). - _Vaclav Kotesovec_, Aug 22 2015
%p A261504 with(numtheory):
%p A261504 a:= n-> `if`(n=0, 1, add(binomial(11*n/d, n/d)
%p A261504              *phi(d), d=divisors(n))/(11*n)):
%p A261504 seq(a(n), n=0..25);
%t A261504 a[n_] := If[n == 0, 1, DivisorSum[n, Binomial[11*n/#, n/#]*EulerPhi[#]&]/ (11*n)];
%t A261504 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 28 2017, translated from Maple *)
%Y A261504 Column k=10 of A261494.
%K A261504 nonn
%O A261504 0,3
%A A261504 _Alois P. Heinz_, Aug 21 2015