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A066318 Number of necklaces with n labeled beads of 2 colors.

%I A066318 #83 Mar 11 2022 07:30:21
%S A066318 2,4,16,96,768,7680,92160,1290240,20643840,371589120,7431782400,
%T A066318 163499212800,3923981107200,102023508787200,2856658246041600,
%U A066318 85699747381248000,2742391916199936000,93241325150797824000,3356687705428721664000,127554132806291423232000
%N A066318 Number of necklaces with n labeled beads of 2 colors.
%C A066318 In the normal probability distribution with mean 0 and standard deviation 1, the expected value E[|x|^(2n-1)] = a(n)/sqrt(2*Pi), while E[|x|^(2n)] = E[x^(2n)] = A001147(n). - _Stanislav Sykora_, Jan 15 2017
%D A066318 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 66 (2.1.27,29).
%H A066318 Vincenzo Librandi, <a href="/A066318/b066318.txt">Table of n, a(n) for n = 1..400</a>
%H A066318 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
%H A066318 Wikipedia, <a href="http://en.wikipedia.org/wiki/Normal_distribution">Normal distribution</a>, formula for E(|x|^p).
%H A066318 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>.
%F A066318 a(n) = (n-1)!*2^n.
%F A066318 E.g.f.: log(1/(1-2*x)).
%F A066318 Let gd(x,n) = (d^n/dx^n)(exp(-(1/2)*x^2)*sqrt(2)/(2*sqrt(Pi))) = (-1)^((1/2)*n)*(x^2)^((1/2)*n)*2^(-(1/2)*n+1/2)*(exp(I*Pi*n)+1)/(4*sqrt(Pi)*GAMMA(1+(1/2)*n)) be the n-th derivative of the standard Gaussian distribution. Evaluating gd(x,n) at x=1 gives gd(1,n) = 2^(-(1/2)*n+1/2)*(exp(I*Pi*n)+1)*(-1)^((1/2)*n)/(4*sqrt(Pi)*GAMMA(1+(1/2)*n)). A066318 is the denominator of the even summands of the Taylor series expansion of the Gaussian distribution evaluated at x=1. a(n)=denom(gd(1, 2*n))/sqrt(Pi). - _Stephen Crowley_, May 16 2009
%F A066318 a(n) = 2*(n-1)*a(n-1). - _R. J. Mathar_, Sep 10 2012
%F A066318 G.f.: G(0), where G(k)= 1  + 1/(1 - 1/(1 + 1/(2*k+2)/x/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 14 2013
%F A066318 a(n) = 2 * (2*n-2)!! = 2 * A000165(n-1). - _Alois P. Heinz_, Jun 22 2017
%F A066318 a(n) = (sqrt(Pi)/Gamma((2*n+3)/2))*Product_{k=0..n-1} binomial(2*(n-k)+1,2). - _Stefano Spezia_, Nov 17 2018
%F A066318 From _Amiram Eldar_, Mar 11 2022: (Start)
%F A066318 Sum_{n>=1} 1/a(n) = sqrt(e)/2 (A019775).
%F A066318 Sum_{n>=1} (-1)^(n+1)/a(n) = 1/(2*sqrt(e)). (End)
%p A066318 with(combstruct):A:=[N,{N=Cycle(Union(Z$2))},labeled]: seq(count(A,size=n),n=1..18); # _Zerinvary Lajos_, Oct 07 2007
%p A066318 # alternative Maple program:
%p A066318 a:= n-> 2*doublefactorial(2*n-2):
%p A066318 seq(a(n), n=1..20);  # _Alois P. Heinz_, Jun 22 2017
%t A066318 mx = 18; Rest[ Range[0, mx]! CoefficientList[ Series[ Log[1/(1 - 2 x)], {x, 0, mx}], x]] (* _Robert G. Wilson v_, Sep 22 2011 *)
%t A066318 Table[(n-1)!*2^n,{n,20}] (* _Harvey P. Dale_, Dec 15 2011 *)
%o A066318 (Magma) [Factorial(n-1)*2^n: n in [1..20]]; // _Vincenzo Librandi_, Sep 23 2011
%o A066318 (PARI) apply( A066318=n->(n-1)!<<n , [1..18]) \\ _M. F. Hasler_, Jan 15 2017
%o A066318 (GAP) a_n:=List([1..10], n->Factorial(n-1)*2^n); # _Stefano Spezia_, Nov 17 2018
%o A066318 (Python) import math
%o A066318 for n in range(1,10): print(math.factorial(n-1)*2**n, end=', ') # _Stefano Spezia_, Nov 17 2018
%o A066318 (Maxima) a(n):=(n-1)!*2^n$ makelist(a(n), n, 1, 10);  /* _Stefano Spezia_, Nov 21 2018 */
%o A066318 (Sage) [2^n*factorial(n-1) for n in (1..20)] # _G. C. Greubel_, Nov 21 2018
%Y A066318 Apart from initial term, same as A032184.
%Y A066318 Cf. A000165, A000796, A001147, A019775, A019727.
%K A066318 nonn
%O A066318 1,1
%A A066318 _Christian G. Bower_, Dec 13 2001