A186229 Expansion of (2F1( (-(1/2), 1/6); (-2/3))( 16 x) -1)/(2*x).

1, 14, 182, 2470, 34580, 494760, 7191690, 105793545, 1570873850, 23500272796, 353724885332, 5351515200668, 81313973049064, 1240116577389200, 18973783634054760, 291115203548084370, 4477664537437798980, 69023046543088792440, 1066084706728274263800, 16495237916832025427160, 255635559046076610807120

Offset: 0

Author

Olivier Gérard, Feb 15 2011

Keywords

• nonn
• easy

Comments

Combinatorial interpretation welcome.

Probably a class of paths (Cf. A135404, A000888)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Mathematica

CoefficientList[Series[(HypergeometricPFQ[{-(1/2), 1/6}, {-(2/3)}, 16 x] - 1)/(2 x), {x, 0, 20}], x]
FullSimplify[Table[-((2^(1/3 + 4 n) (-(4/3))! (-(1/2) + n)! (1/6 + n)!)/(Pi (-(2/3) + n)! (1 + n)!)), {n, 0, 20}]] (* Benedict W. J. Irwin, Jul 12 2016 *)

Formula

D-finite with recurrence (n+1)*(3n-2)*a(n) = 4*(6n+1)*(2n-1)*a(n-1). - R. J. Mathar, Jul 11 2012

a(n) ~ 3*GAMMA(2/3)*2^(1/3) * 16^n/(Pi*n^(2/3)). - Vaclav Kotesovec, Aug 13 2013

a(n) = -2^(1/3+4*n)*(-4/3)!*(-1/2+n)!*(1/6+n)!/(Pi*(-2/3+n)!*(1+n)!). - Benedict W. J. Irwin, Jul 12 2016