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%I #63 Jul 28 2023 18:57:06
%S 2,2,6,30,210,1890,20790,270270,4054050,68918850,1309458150,
%T 27498621150,632468286450,15811707161250,426916093353750,
%U 12380566707258750,383797567925021250,12665319741525701250,443286190953399543750,16401589065275783118750,639661973545755541631250
%N a(n) = (2*n)!/(n!*2^(n-1)).
%C Right-hand edge of triangle in A097749.
%C 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
%H Muniru A Asiru, <a href="/A097801/b097801.txt">Table of n, a(n) for n = 0..150</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%F a(n) = 2*A001147(n).
%F a(n) = 2*(2*n-1)!!. - _Johannes W. Meijer_, Nov 12 2009
%F E.g.f.: 2/sqrt(1 - 2*x). - _Sergei N. Gladkovskii_, Jul 06 2012
%F 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
%F a(n) = 2 * Product_{i=1..n} denominator(i!/(2*i - 1)). - _Wesley Ivan Hurt_, Oct 12 2013
%F D-finite with recurrence: a(n) + (-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 20 2020
%F From _Amiram Eldar_, Feb 24 2021: (Start)
%F Sum_{n>=0} 1/a(n) = 1/2 + sqrt(e*Pi/2)*erf(1/sqrt(2))/2, where erf(x) is the error function.
%F 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)
%p a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(2*n-1)) end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 27 2013
%t FoldList[Times, 2, Range[1, 39, 2]] (* _Arkadiusz Wesolowski_, May 08 2012 *)
%t 2(2*Range[0,20]-1)!! (* _Harvey P. Dale_, Apr 22 2013 *)
%t CoefficientList[Series[2/Sqrt[1-2*x], {x, 0, 45}], x]*Table[k !, {k, 0, 45}] (* _Stefano Spezia_, Sep 04 2018 *)
%o (Magma) [Factorial(2*n)/(Factorial(n)*2^(n-1)): n in [0..20]]; // _Vincenzo Librandi_, Aug 21 2018
%o (GAP) List([0..20],n->Factorial(2*n)/(Factorial(n)*2^(n-1))); # _Muniru A Asiru_, Aug 21 2018
%o (PARI) x='x+O('x^30); Vec(serlaplace(2*(1-2*x)^(-1/2))) \\ _Altug Alkan_, Sep 05 2018
%Y Cf. A001147, A097749.
%K nonn
%O 0,1
%A _N. J. A. Sloane_, Sep 21 2004
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