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a(n) = ((2*n)!/n!)*2^(2*n+1).
0

%I #34 Jun 03 2022 18:17:41

%S 0,2,16,384,15360,860160,61931520,5449973760,566797271040,

%T 68015672524800,9250131463372800,1406019982432665600,

%U 236211357048687820800,43462889696958559027200,8692577939391711805440000,1877596834908609749975040000,435602465698797461994209280000

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

%C A simple context-free grammar in a labeled universe.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=693">Encyclopedia of Combinatorial Structures 693</a>.

%F E.g.f.: 1/4 - (1/4)*sqrt(1-16*x).

%F D-finite Recurrence: {a(1)=2, (8-16*n)*a(n) + a(n+1)=0}.

%F a(n) = (1/8)*16^(n+1)*Gamma(n+1/2)/Pi^(1/2).

%F a(n) = n! * A052707(n). - _R. J. Mathar_, Aug 21 2014

%F From _Amiram Eldar_, Mar 22 2022: (Start)

%F Sum_{n>=1} 1/a(n) = sqrt(Pi)*exp(1/16)*erf(1/4)/8, where erf is the error function.

%F 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)

%p 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 [seq((2*n)!/n!*2^(2*n+1), n=0..12)]; # _Zerinvary Lajos_, Sep 28 2006

%t 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 Cf. A052707.

%K easy,nonn

%O 0,2

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E Better definition from _Zerinvary Lajos_, Sep 28 2006

%E More terms from _Harvey P. Dale_, Aug 12 2015