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A066318
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Number of necklaces with n labeled beads of 2 colors.
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7
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2, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400, 163499212800, 3923981107200, 102023508787200, 2856658246041600, 85699747381248000, 2742391916199936000, 93241325150797824000, 3356687705428721664000, 127554132806291423232000
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 66 (2.1.27,29).
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LINKS
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FORMULA
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a(n) = (n-1)!*2^n.
E.g.f.: log(1/(1-2*x)).
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
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
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
Sum_{n>=1} 1/a(n) = sqrt(e)/2 (A019775).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/(2*sqrt(e)). (End)
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MAPLE
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with(combstruct):A:=[N, {N=Cycle(Union(Z$2))}, labeled]: seq(count(A, size=n), n=1..18); # Zerinvary Lajos, Oct 07 2007
# alternative Maple program:
a:= n-> 2*doublefactorial(2*n-2):
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MATHEMATICA
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mx = 18; Rest[ Range[0, mx]! CoefficientList[ Series[ Log[1/(1 - 2 x)], {x, 0, mx}], x]] (* Robert G. Wilson v, Sep 22 2011 *)
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PROG
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(GAP) a_n:=List([1..10], n->Factorial(n-1)*2^n); # Stefano Spezia, Nov 17 2018
(Python) import math
for n in range(1, 10): print(math.factorial(n-1)*2**n, end=', ') # Stefano Spezia, Nov 17 2018
(Maxima) a(n):=(n-1)!*2^n$ makelist(a(n), n, 1, 10); /* Stefano Spezia, Nov 21 2018 */
(Sage) [2^n*factorial(n-1) for n in (1..20)] # G. C. Greubel, Nov 21 2018
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CROSSREFS
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Apart from initial term, same as A032184.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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