This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A032194 #42 Mar 04 2025 07:54:23
%S A032194 1,1,5,19,55,143,335,715,1430,2704,4862,8398,14000,22610,35530,54484,
%T A032194 81719,120175,173593,246675,345345,476913,650325,876525,1168710,
%U A032194 1542684,2017356,2615104,3362260,4289780,5433736,6835972
%N A032194 Number of necklaces with 9 black beads and n-9 white beads.
%C A032194 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 A032194 Christian G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032194 David Broadhurst and Xavier Roulleau, <a href="https://arxiv.org/abs/2502.19523">Number of partitions of modular integers</a>, arXiv:2502.19523 [math.NT], 2025. See p. 19.
%H A032194 Frank Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A032194 Frank Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A032194 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H A032194 <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,22,-27,36,-42,36,-27,23,-21,21,-23,27,-36,42,-36,27,-22,15,-6,1).
%F A032194 "CIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...
%F A032194 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 A032194 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 A032194 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 A032194 Column k=9 of A047996.
%Y A032194 Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193.
%K A032194 nonn
%O A032194 9,3
%A A032194 _Christian G. Bower_