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Number of 2n-bead black-white strings with n black beads and fundamental period 2n.
14

%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