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A052737
a(n) = ((2*n)!/n!)*2^(2*n+1).
0
0, 2, 16, 384, 15360, 860160, 61931520, 5449973760, 566797271040, 68015672524800, 9250131463372800, 1406019982432665600, 236211357048687820800, 43462889696958559027200, 8692577939391711805440000, 1877596834908609749975040000, 435602465698797461994209280000
OFFSET
0,2
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.
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
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 *)
CROSSREFS
Cf. A052707.
Sequence in context: A325287 A140308 A280723 * A002474 A375059 A172149
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Better definition from Zerinvary Lajos, Sep 28 2006
More terms from Harvey P. Dale, Aug 12 2015
STATUS
approved