A383937 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(2/3).

1, 6, 33, 180, 990, 5508, 30978, 175824, 1005345, 5782590, 33418737, 193876092, 1128297276, 6583492080, 38498441400, 225550220544, 1323563204394, 7777806812892, 45762197971050, 269545947941160, 1589219394582996, 9378142402189176, 55385341859409948

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. A016137, A383935.

Programs

Magma

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

Mathematica

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

PARI

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

Formula

G.f.: B(x)^(2/3), where B(x) is the g.f. of A016137.

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

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

D-finite with recurrence n*a(n) +3*(-3*n+1)*a(n-1) +6*(3*n-2)*a(n-2)=0. - R. J. Mathar, Aug 26 2025