A185047 Expansion of 2F1( [1, 4/3]; [3]; 9*x).

1, 4, 21, 126, 819, 5616, 40014, 293436, 2200770, 16805880, 130245570, 1021926780, 8102419470, 64819355760, 522606055815, 4242331511910, 34645707347265, 284459491903860, 2346790808206845, 19444838125142430, 161745698950048395, 1350224965148230080

Offset

0,2

Comments

Close to A003168.

Can be seen as a degree 3 analog of the Catalan numbers A000108 (which would be degree 2).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = -9^(n+1)*binomial(n+1/3, n+2). - Karol A. Penson, Nov 06 2015

a(n) = (1/(6*sqrt(3)*Pi))*Integral_{x = 0..9} x^n*x^(1/3)*(9 - x)^(2/3). Cf. A034164. - Peter Bala, Nov 17 2015

O.g.f.: (1 - (1-9*x)^(2/3) - 6*x)/(9*x^2).

D-finite with recurrence (n+2)*a(n) +3*(-3*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

Sum_{n>=0} 1/a(n) = (3/512)*(92 + 15*Pi*sqrt(3) + 45*log(3)). - Amiram Eldar, Dec 18 2022

a(n) = ((n + 3)/3) * Product_{1 <= i <= j <= n} (2*i + j + 3)/(2*i + j - 1). - Peter Bala, Feb 22 2023

Maple

A185047:=n->-9^(n+1)*binomial(n+1/3, n+2): seq(A185047(n), n=0..30); # Wesley Ivan Hurt, Feb 16 2017

Mathematica

CoefficientList[Series[ HypergeometricPFQ[{1, 4/3}, {3}, 9 x], {x, 0, 20}], x]

Crossrefs

Cf. A000108, A003168, A034164.

Sequence in context: A275758 A003168 A211249 * A353608 A032326 A281581

Adjacent sequences: A185044 A185045 A185046 * A185048 A185049 A185050

Keyword

nonn,easy

Author

Olivier Gérard, Feb 15 2011

Status

approved