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A032194 Number of necklaces with 9 black beads and n-9 white beads. 2

%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_

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