A180400 Coefficients of Maclaurin series for (1-9x-9x^2)^(-1/3).

1, 3, 21, 162, 1341, 11529, 101619, 911466, 8281737, 76002381, 703017549, 6544803564, 61254970686, 575885086182, 5434948357146, 51462813578148, 488705091057981, 4652700300002475, 44395945025504625, 424479488258350350

Offset: 0

Author

Clark Kimberling, Sep 01 2010, Sep 02 2010

Keywords

• nonn
• easy

Examples

			The Maclaurin series begins with 1 + 3x + 21x^2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A180399.

Programs

Mathematica

CoefficientList[Series[Power[1-9x-9x^2, (-3)^-1],{x,0,20}],x] (* Harvey P. Dale, Mar 11 2012 *)

Formula

G.f.: (1-9x-9x^2)^(-1/3).

D-finite with recurrence: n*a(n) = 3*(3*n-2)*a(n-1) + 3*(3*n-4)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ sqrt(3)*Gamma(2/3)/(2^(2/3)*(13-3*sqrt(13))^(1/3)*Pi) * ((9+3*sqrt(13))/2)^n/(n^(2/3)). - Vaclav Kotesovec, Oct 20 2012

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(k,n-k). - Seiichi Manyama, Mar 27 2023