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A052737 a(n) = ((2*n)!/n!)*2^(2*n+1).

%I A052737 #44 Apr 30 2025 11:09:19
%S A052737 0,2,16,384,15360,860160,61931520,5449973760,566797271040,
%T A052737 68015672524800,9250131463372800,1406019982432665600,
%U A052737 236211357048687820800,43462889696958559027200,8692577939391711805440000,1877596834908609749975040000,435602465698797461994209280000
%N A052737 a(n) = ((2*n)!/n!)*2^(2*n+1).
%C A052737 A simple context-free grammar in a labeled universe.
%H A052737 Tianji Cai, François Charton, Kyle Cranmer, Lance J. Dixon, Garrett W. Merz, and Matthias Wilhelm, <a href="https://arxiv.org/abs/2501.05743">Recurrent Features of Amplitudes in Planar N = 4 Super Yang-Mills Theory</a>, arXiv:2501.05743 [hep-th], 2025. See pp. 12, 29. See also <a href="https://doi.org/10.1007/JHEP04(2025)143">J. High Energy Phys.</a>, (JHEP 2025) Vol. 2025, Art. No. 143. See p. 9.
%H A052737 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=693">Encyclopedia of Combinatorial Structures 693</a>.
%F A052737 E.g.f.: 1/4 - (1/4)*sqrt(1-16*x).
%F A052737 D-finite Recurrence: a(1)=2, (8-16*n)*a(n) + a(n+1)=0, i.e. a(n) +8*(-2*n+3)*a(n-1)=0.
%F A052737 a(n) = (1/8)*16^(n+1)*Gamma(n+1/2)/Pi^(1/2).
%F A052737 a(n) = n! * A052707(n). - _R. J. Mathar_, Aug 21 2014
%F A052737 From _Amiram Eldar_, Mar 22 2022: (Start)
%F A052737 Sum_{n>=1} 1/a(n) = sqrt(Pi)*exp(1/16)*erf(1/4)/8, where erf is the error function.
%F A052737 Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(Pi)*exp(-1/16)*erfi(1/4)/8, where erfi is the imaginary error function. (End)
%F A052737 a(n)=2*A052734(n). - _R. J. Mathar_, Jan 13 2025
%p A052737 spec := [S,{B=Union(Z,C),S=Union(B,Z,C),C=Prod(S,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052737 [seq((2*n)!/n!*2^(2*n+1), n=0..12)]; # _Zerinvary Lajos_, Sep 28 2006
%t A052737 With[{nn=20},CoefficientList[Series[1/4-Sqrt[1-16x]/4,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 12 2015 *)
%Y A052737 Cf. A052707.
%K A052737 easy,nonn
%O A052737 0,2
%A A052737 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052737 Better definition from _Zerinvary Lajos_, Sep 28 2006
%E A052737 More terms from _Harvey P. Dale_, Aug 12 2015