A254282 Expansion of (1 - (1 - 27*x)^(1/3)) / (9*x).

1, 9, 135, 2430, 48114, 1010394, 22084326, 496897335, 11428638705, 267430145697, 6345388002447, 152289312058728, 3690087176807640, 90143558176300920, 2217531531137002632, 54883905395640815142, 1365640704844474400298, 34141017621111860007450

Offset: 0

Author

Vaclav Kotesovec, Jan 27 2015

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..700

Crossrefs

Cf. A000108, A008544, A254286, A254287, A025748.

Programs

Magma

[Round(3^(3*n)*Gamma(n+2/3)/(Gamma(2/3)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022

Mathematica

CoefficientList[Series[(1-(1-27*x)^(1/3))/(9*x),{x,0,20}],x]
CoefficientList[Series[Hypergeometric1F1[2/3,2,27*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
nxt[{n_,a_}]:={n+1,((27n+18)*a)/(n+2)}; NestList[nxt,{0,1},20][[All,2]] (* Harvey P. Dale, Jun 03 2019 *)

SageMath

[3^(3*n)*rising_factorial(2/3,n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

Formula

G.f.: (1 - (1-27*x)^(1/3)) / (9*x).

a(n) ~ 3^(3*n) / (Gamma(2/3) * n^(4/3)).

Recurrence: (n+1)*a(n) = 9*(3*n-1)*a(n-1).

a(n) = 27^n * Gamma(n+2/3) / (Gamma(2/3) * Gamma(n+2)).

E.g.f.: hypergeom([2/3], [2], 27*x). - Vaclav Kotesovec, Jan 28 2015

From Peter Bala, Sep 01 2017: (Start)

a(n) = (-1)^n*binomial(1/3, n+1)*3^(3*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).

a(n) = 3^n*A025748(n+1). (End)