Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #37 Apr 30 2019 11:21:52
%S 1,1,5,19,55,143,335,715,1430,2704,4862,8398,14000,22610,35530,54484,
%T 81719,120175,173593,246675,345345,476913,650325,876525,1168710,
%U 1542684,2017356,2615104,3362260,4289780,5433736,6835972
%N Number of necklaces with 9 black beads and n-9 white beads.
%C The g.f. is Z(C_9,x)/x^9, the 9-variate cycle index polynomial for the cyclic group C_9, with substitution x[i]->1/(1-x^i), i=1,...,9. Therefore by Polya enumeration a(n+9) is the number of cyclically inequivalent 9-necklaces whose 9 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_9,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - _Wolfdieter Lang_, Feb 15 2005
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</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 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F "CIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...
%F G.f.: (x^9)*(1-5*x+14*x^2-18*x^3+21*x^4-21*x^5+25*x^6 -21*x^7 +21*x^8 -18*x^9 +14*x^10 -5*x^11 +x^12) / ((1-x)^6*(1-x^3)^2*(1-x^9)).
%F G.f.: (1/9)*x^9*(1/(1-x)^9+2/(1-x^3)^3+6/(1-x^9)^1). - _Herbert Kociemba_, Oct 22 2016
%t k = 9; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* _Robert A. Russell_, Sep 27 2004 *)
%Y Column k=9 of A047996.
%Y Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193.
%K nonn
%O 9,3
%A _Christian G. Bower_