A101601 G.f.: c(3x)^3, c(x) 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

Author

Paul Barry, Dec 08 2004

Keywords

• easy
• nonn

Crossrefs

Cf. A050159, A101600, A101602.

Programs

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 *)

Formula

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

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

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)