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A066318
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Number of necklaces with n labeled beads of 2 colors.
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4
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2, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400, 163499212800, 3923981107200, 102023508787200, 2856658246041600, 85699747381248000, 2742391916199936000, 93241325150797824000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pg 66 (2.1.27,29).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..400
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for sequences related to necklaces
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FORMULA
| a(n)=(n-1)!*2^n. E.g.f.: log(1/(1-2x)).
Let gd(x,n)=Diff(exp(-(1/2)*x^2)*sqrt(2)/(2*sqrt(Pi)), x$n)]=(-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. A066318[n]=denom(gd(1, 2*n))/sqrt(Pi) [From Stephen Crowley, May 16 2009]
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MAPLE
| seq(count(Permutation(n))*count(Subset(n+1)), n=0..17); - Zerinvary Lajos, Oct 16 2006
with(combstruct):A:=[N, {N=Cycle(Union(Z$2))}, labeled]: seq(count(A, size=n), n=1..18); - Zerinvary Lajos, Oct 07 2007
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MATHEMATICA
| mx = 18; Rest[ Range[0, mx]! CoefficientList[ Series[ Log[1/(1 - 2 x)], {x, 0, mx}], x]] (* Robert G. Wilson v, Sept 22 2011 *)
Table[(n-1)!*2^n, {n, 20}] (* From Harvey P. Dale, Dec 15 2011 *)
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PROG
| (MAGMA) [Factorial(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Sep 23 2011
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CROSSREFS
| Apart from initial term, same as A032184.
Sequence in context: A009565 A009838 A088335 * A066952 A135249 A110365
Adjacent sequences: A066315 A066316 A066317 * A066319 A066320 A066321
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Dec 13 2001
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EXTENSIONS
| Added formula involving Gaussian series expansion [From Stephen Crowley, May 16 2009]
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