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A005648 Number of 2n-bead black-white reversible necklaces with n black beads.

%I A005648 M0878 #68 Jul 03 2020 06:23:11
%S A005648 1,1,2,3,8,16,50,133,440,1387,4752,16159,56822,200474,718146,2587018,
%T A005648 9398520,34324174,126068558,465093571,1723176308,6407924300,
%U A005648 23910576230,89494164973,335913918902,1264107416466
%N A005648 Number of 2n-bead black-white reversible necklaces with n black beads.
%C A005648 a(n) is the coefficient of c_1^n*c_2^n in the cycle index polynomial for the dihedral group D_{2*n} evaluated with the figure counting polynomial c = c_1 + c_2, n>=1, abbreviated as Z(D_{2*n},c). See, e.g., the Harary-Palmer reference (given under A212355), p. 42, Theorem (PET), and the example for all 6 two-colored 4-bracelets (called there necklaces) on p. 44, Figure 2.4.2. - _Wolfdieter Lang_, Jun 05 2012
%D A005648 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005648 G. C. Greubel, <a href="/A005648/b005648.txt">Table of n, a(n) for n = 0..1665</a> (terms 0..200 from Andrew Howroyd)
%H A005648 Marcia Ascher, <a href="http://www.jstor.org/stable/2690304">Mu torere: an analysis of a Maori game</a>, Math. Mag. 60 (1987), no. 2, 90-100.
%H A005648 R. K. Guy & N. J. A. Sloane, <a href="/A005648/a005648.pdf">Correspondence, 1985</a>
%H A005648 Paul Melotti, Sanjay Ramassamy, Paul Thévenin, <a href="https://arxiv.org/abs/2003.11006">Points and lines configurations for perpendicular bisectors of convex cyclic polygons</a>, arXiv:2003.11006 [math.CO], 2020.
%H A005648 E. M. Palmer and R. W. Robinson, <a href="http://projecteuclid.org/euclid.pjm/1102711113">Enumeration of self-dual configurations</a> Pacific J. Math., 110 (1984), 203-221.
%H A005648 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A005648 F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A005648 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F A005648 a(n) = ( Sum_{d|n} phi(n/d)*C(2*d, d) )/(4*n) + C(2*k, k)/2, where k = floor(n/2). - _Michael Somos_
%F A005648 a(n) = (A003239(n) + C(2*k, k))/2, where k = [ n/2 ]. - R. J. Fletcher, (yylee(AT)mail.ncku.edu.tw)
%e A005648 a(2) = 2: BBWW, BWBW.
%e A005648 a(3) = 3: BBBWWW, BBWBWW, BWBWBW.
%e A005648 a(4) = 8: BBBBWWWW, BBBWBWWW, BBBWWBWW, BBWWBBWW, BBWBWBWW, BBWBWWBW, BBWBBWWW, BWBWBWBW.
%t A005648 f[k_Integer, n_] := (Plus @@ (EulerPhi[ # ]Binomial[n/#, k/# ] & /@ Divisors[GCD[n, k]])/n + Binomial[(n - If[OddQ@n, 1, If[OddQ@k, 2, 0]])/2, (k - If[OddQ@k, 1, 0])/2])/2 (* _Robert A. Russell_, Sep 27 2004 *)
%t A005648 Table[ f[n, 2n], {n, 27}] (* _Robert G. Wilson v_, Mar 29 2006 *)
%t A005648 a[0] = 1; a[n_] := 1/2*(Binomial[2*Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#]*Binomial[2*n/#, n/#]&]/(2*n)); Array[a, 26, 0] (* _Jean-François Alcover_, Nov 05 2017, translated from PARI *)
%o A005648 (PARI) a(n) = 1/2*( binomial(2*(n\2), n\2) + if(n<1, n >= 0, sumdiv(n, k, eulerphi(k)*binomial(2*n/k, n/k))/(2*n) ));
%Y A005648 Cf. A000984, A003239.
%K A005648 nonn,nice,easy
%O A005648 0,3
%A A005648 _N. J. A. Sloane_
%E A005648 Sequence extended and description corrected by _Christian G. Bower_
%E A005648 Example n=8 (word no. 6) corrected by _Wolfdieter Lang_, Jun 05 2012