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

0, 2, 16, 384, 15360, 860160, 61931520, 5449973760, 566797271040, 68015672524800, 9250131463372800, 1406019982432665600, 236211357048687820800, 43462889696958559027200, 8692577939391711805440000, 1877596834908609749975040000, 435602465698797461994209280000

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

A simple context-free grammar in a labeled universe.

Links

• Tianji Cai, François Charton, Kyle Cranmer, Lance J. Dixon, Garrett W. Merz, and Matthias Wilhelm, Recurrent Features of Amplitudes in Planar N = 4 Super Yang-Mills Theory, arXiv:2501.05743 [hep-th], 2025. See pp. 12, 29. See also J. High Energy Phys., (JHEP 2025) Vol. 2025, Art. No. 143. See p. 9.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 693.

Crossrefs

Cf. A052707.

Programs

Maple

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);
[seq((2*n)!/n!*2^(2*n+1), n=0..12)]; # Zerinvary Lajos, Sep 28 2006

Mathematica

With[{nn=20},CoefficientList[Series[1/4-Sqrt[1-16x]/4,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2015 *)

Formula

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

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.

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

a(n) = n! * A052707(n). - R. J. Mathar, Aug 21 2014

From Amiram Eldar, Mar 22 2022: (Start)

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

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)

a(n)=2*A052734(n). - R. J. Mathar, Jan 13 2025

Extensions

Better definition from Zerinvary Lajos, Sep 28 2006

More terms from Harvey P. Dale, Aug 12 2015