A180399 Expansion of (1/3)*(1 - (1-9*x-9*x^2)^(1/3)).

0, 1, 4, 21, 138, 999, 7683, 61542, 507663, 4281849, 36748998, 319845591, 2816007714, 25032803841, 224355173193, 2024955168606, 18388543939947, 167882583075453, 1540000362501702, 14186252492098011, 131176523761136568, 1217094112710349731, 11327464549934673309

Offset: 0

Author

Clark Kimberling, Sep 01 2010

Keywords

• nonn

Examples

			The Maclaurin series begins with x + 4x^2 + 21x^3.
		

Links

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

Crossrefs

Cf. A180400.

Programs

Mathematica

CoefficientList[Series[1/3*(1-(1-9*x-9*x^2)^(1/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

x='x+O('x^66); concat([0],Vec(1/3*(1-(1-9*x-9*x^2)^(1/3)))) \\ Joerg Arndt, Jun 01 2013

Formula

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

a(n) = sum(m=1..n, binomial(m,n-m)/m * sum(k=0..m-1, binomial(k,m-1-k) * 3^k*(-1)^(m-1-k) * binomial(m+k-1,m-1))). [From Vladimir Kruchinin, Feb 08 2011]

Recurrence: n*a(n) = 3*(3*n-4)*a(n-1) + 3*(3*n-8)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ ((13-3*sqrt(13))/2)^(1/3)/(9*Gamma(2/3)) * ((9+3*sqrt(13))/2)^n/(n^(4/3)). - Vaclav Kotesovec, Oct 20 2012