A101601 Expansion of g.f. c(3x)^3, where c(x) is the g.f. of A000108.

1, 9, 81, 756, 7290, 72171, 729729, 7505784, 78298974, 826489170, 8811646074, 94753804536, 1026499549140, 11192793160815, 122744496427425, 1352917116177840, 14979996753469110, 166542316847391870, 1858400773709785470, 20806975169765062200, 233671377667405024620

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f.: 8/(1+sqrt(1-12*x))^3.

a(n) = (3*n+3)/(n+3) * 3^n * C(n+1), where C(n) = A000108(n).

Conjecture: (n+3)*a(n) - 3*(5*n+7)*a(n-1) + 18*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 15 2011

From Amiram Eldar, May 15 2022: (Start)

Sum_{n>=0} 1/a(n) = 51/121 + 964*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).

Sum_{n>=0} (-1)^n/a(n) = 57/169 + 1204*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)). (End)

a(n) ~ 12^(n+1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 10 2025

Mathematica

terms = 18;
c[x_] = (1 - Sqrt[1 - 4x])/(2x) + O[x]^terms // Normal;
CoefficientList[c[3x]^3, x][[1 ;; terms]] (* Jean-François Alcover, Dec 15 2017 *)

Crossrefs

Cf. A000108, A050159, A101600, A101602.

Sequence in context: A061587 A033145 A158762 * A233737 A344298 A144821

Adjacent sequences: A101598 A101599 A101600 * A101602 A101603 A101604

Keyword

easy,nonn

Author

Paul Barry, Dec 08 2004

Status

approved