%I #24 May 05 2023 10:59:55
%S 1,2,4,18,64,250,900,3430,12800,48600,184500,705430,2703168,10400598,
%T 40113164,155117250,601067520,2333606218,9075085776,35345263798,
%U 137846344000,538257870990,2104098258284,8233430727598,32247600966144
%N Number of 2n-bead black-white strings with n black beads and fundamental period 2n.
%C For n>0, a(n) is divisible by n^2 (cf. A268619) and 6*a(n) is divisible by n^3 (cf. A268592). - _Max Alekseyev_, Feb 07 2016
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%F For n>0, a(n) = Sum_{d|n} A008683(n/d)*A000984(d).
%F For n>0, a(n) = 2 * A045630(n).
%F a(0)=1, a(n) = n * A060165(n) = 2n * A022553(n). - _Ralf Stephan_, Sep 01 2003
%p A007727 := proc(n)
%p if n = 0 then
%p 1;
%p else
%p add(numtheory[mobius](n/d)*binomial(2*d,d), d =numtheory[divisors](n)) ;
%p end if ;
%p end proc:
%p seq(A007727(n),n=0..10) ; # _R. J. Mathar_, Nov 10 2021
%t a[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]];
%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, May 05 2023 *)
%o (PARI) { a(n) = if(n>0,sumdiv(n, d, moebius(n/d)*binomial(2*d, d)),0); }
%Y Cf. A045630, A060165, A022553.
%K nonn
%O 0,2
%A Doug Bowman, bowman(AT)math.uiuc.edu.
%E Edited by _Max Alekseyev_, Feb 09 2016