%I #28 Mar 08 2024 11:59:40

%S 1,9,135,2430,48114,1010394,22084326,496897335,11428638705,

%T 267430145697,6345388002447,152289312058728,3690087176807640,

%U 90143558176300920,2217531531137002632,54883905395640815142,1365640704844474400298,34141017621111860007450

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

%H Harvey P. Dale, <a href="/A254282/b254282.txt">Table of n, a(n) for n = 0..700</a>

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

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

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

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

%F E.g.f.: hypergeom([2/3], [2], 27*x). - _Vaclav Kotesovec_, Jan 28 2015

%F From _Peter Bala_, Sep 01 2017: (Start)

%F 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).

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

%t CoefficientList[Series[(1-(1-27*x)^(1/3))/(9*x),{x,0,20}],x]

%t CoefficientList[Series[Hypergeometric1F1[2/3,2,27*x],{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2015 *)

%t nxt[{n_,a_}]:={n+1,((27n+18)*a)/(n+2)}; NestList[nxt,{0,1},20][[All,2]] (* _Harvey P. Dale_, Jun 03 2019 *)

%o (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

%o (SageMath) [3^(3*n)*rising_factorial(2/3,n)/factorial(n+1) for n in (0..30)] # _G. C. Greubel_, Aug 10 2022

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

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Jan 27 2015