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A097801
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a(n) = (2*n)!/(n!*2^(n-1)).
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12
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2, 2, 6, 30, 210, 1890, 20790, 270270, 4054050, 68918850, 1309458150, 27498621150, 632468286450, 15811707161250, 426916093353750, 12380566707258750, 383797567925021250, 12665319741525701250, 443286190953399543750, 16401589065275783118750, 639661973545755541631250
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OFFSET
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0,1
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COMMENTS
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Right-hand edge of triangle in A097749.
Also, the number of ways to paint the 2*n cells of dimension n - 1 that bound a regular convex n-cube polytope using exactly 2n colors where n > 0 is the dimension of Euclidean space. - Frank M Jackson, Aug 13 2018
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LINKS
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FORMULA
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G.f.: G(0), where G(k)= 1 + 1/(1 - x*(2*k + 1)/(x*(2*k + 1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = 2 * Product_{i=1..n} denominator(i!/(2*i - 1)). - Wesley Ivan Hurt, Oct 12 2013
D-finite with recurrence: a(n) + (-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1/2 + sqrt(e*Pi/2)*erf(1/sqrt(2))/2, where erf(x) is the error function.
Sum_{n>=0} (-1)^n/a(n) = 1/2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2))/2, where erfi(x) is the imaginary error function. (End)
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MAPLE
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a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(2*n-1)) end:
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MATHEMATICA
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CoefficientList[Series[2/Sqrt[1-2*x], {x, 0, 45}], x]*Table[k !, {k, 0, 45}] (* Stefano Spezia, Sep 04 2018 *)
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PROG
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(Magma) [Factorial(2*n)/(Factorial(n)*2^(n-1)): n in [0..20]]; // Vincenzo Librandi, Aug 21 2018
(GAP) List([0..20], n->Factorial(2*n)/(Factorial(n)*2^(n-1))); # Muniru A Asiru, Aug 21 2018
(PARI) x='x+O('x^30); Vec(serlaplace(2*(1-2*x)^(-1/2))) \\ Altug Alkan, Sep 05 2018
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CROSSREFS
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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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