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A007727 Number of 2n-bead black-white strings with n black beads and fundamental period 2n.

%I A007727 #26 Jul 08 2025 01:01:48
%S A007727 1,2,4,18,64,250,900,3430,12800,48600,184500,705430,2703168,10400598,
%T A007727 40113164,155117250,601067520,2333606218,9075085776,35345263798,
%U A007727 137846344000,538257870990,2104098258284,8233430727598,32247600966144
%N A007727 Number of 2n-bead black-white strings with n black beads and fundamental period 2n.
%C A007727 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 A007727 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 A007727 For n>0, a(n) = Sum_{d|n} A008683(n/d)*A000984(d).
%F A007727 For n>0, a(n) = 2 * A045630(n).
%F A007727 a(0)=1, a(n) = n * A060165(n) = 2n * A022553(n). - _Ralf Stephan_, Sep 01 2003
%p A007727 A007727 := proc(n)
%p A007727     if n = 0 then
%p A007727         1;
%p A007727     else
%p A007727         add(numtheory[mobius](n/d)*binomial(2*d,d), d =numtheory[divisors](n)) ;
%p A007727     end if ;
%p A007727 end proc:
%p A007727 seq(A007727(n),n=0..10) ; # _R. J. Mathar_, Nov 10 2021
%t A007727 a[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]];
%t A007727 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, May 05 2023 *)
%o A007727 (PARI) { a(n) = if(n>0,sumdiv(n, d, moebius(n/d)*binomial(2*d, d)),0); }
%Y A007727 Cf. A045630, A060165, A022553.
%K A007727 nonn
%O A007727 0,2
%A A007727 Doug Bowman, bowman(AT)math.uiuc.edu
%E A007727 Edited by _Max Alekseyev_, Feb 09 2016