A383936 Expansion of 1 / ( (1-6*x) * (1+3*x)^2 )^(1/3).

1, 0, 9, 18, 162, 648, 4050, 20412, 117369, 639576, 3628233, 20360970, 115858512, 659144304, 3772679976, 21637128240, 124518397770, 718139766240, 4151542099770, 24045292507860, 139520453553468, 810834881716080, 4719151317787452, 27502373918849544

Offset: 0

Author

Seiichi Manyama, Aug 18 2025

Keywords

• nonn
• easy
• changed

Links

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

Crossrefs

Cf. A383935.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-6*x) * (1+3*x)^2 )^(1/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

Mathematica

CoefficientList[Series[1/((1-6*x)*(1+3*x)^2)^(1/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

PARI

a(n) = (-3)^n*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(-1/3, k)*binomial(-2/3, n-k));

Formula

a(n) = (-3)^n * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(-1/3,k) * binomial(-2/3,n-k).

a(n) ~ 2^(n + 2/3) * 3^(n - 2/3) / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 18 2025

D-finite with recurrence n*a(n) +3*(-n+1)*a(n-1) +18*(-n+1)*a(n-2)=0. - R. J. Mathar, Aug 26 2025