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A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.

%I A189980 #28 Feb 07 2021 00:51:12
%S A189980 1,1,6,6,21,21,56,56,126,126,252,252,462,462,792,792,1287,1287,2002,
%T A189980 2002,3003,3003,4368,4368,6188,6188,8568,8568,11628,11628,15504,15504,
%U A189980 20349,20349,26334,26334,33649,33649
%N A189980 a(n) is the number of incongruent two-color bracelets of n beads, 10 from them are black (A005515), having a diameter of symmetry.
%C A189980 For n >= 11, a(n-1) is the number of incongruent two-color bracelets of n beads, 11 from them are black (A032282), having a diameter of symmetry.
%H A189980 Hansraj Gupta, <a href="https://web.archive.org/web/20200806162943/https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a66_964.pdf">Enumeration of incongruent cyclic k-gons</a>, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
%H A189980 V. Shevelev, <a href="http://arxiv.org/abs/0710.1370">A problem of enumeration of two-color bracelets with several variations</a>, arXiv:0710.1370 [math.CO], 2007-2011.
%H A189980 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).
%F A189980 a(n) = binomial(floor(n/2), 5). [Typo fixed by _Colin Barker_, Feb 07 2013]
%F A189980 a(n+6) = A194005(n, n-5). - _Johannes W. Meijer_, Aug 15 2011
%F A189980 G.f.: x^10/((x-1)^6*(x+1)^5). - _Colin Barker_, Feb 07 2013
%p A189980 A189980 :=proc(n): binomial(floor(n/2),5) end: seq(A189980(n), n=10..47); # _Johannes W. Meijer_, Aug 15 2011
%t A189980 Table[Binomial[Floor[n/2],5],{n,10,50}] (* _Harvey P. Dale_, Oct 06 2017 *)
%Y A189980 Cf. A005515, A032282, A008805, A058187, A189976.
%K A189980 nonn,easy
%O A189980 10,3
%A A189980 _Vladimir Shevelev_, May 03 2011
%E A189980 Data added and link corrected by _Johannes W. Meijer_, Aug 15 2011