A080421 a(n) = (n+1)*(n+2)*(n+9)*3^n/18.

1, 10, 66, 360, 1755, 7938, 34020, 139968, 557685, 2165130, 8227494, 30705480, 112842639, 409209570, 1466777160, 5203870272, 18294856425, 63795240522, 220829678730, 759344158440, 2595329855811, 8821564534530, 29832927334956, 100419390748800, 336561864306525

Offset: 0

Author

Paul Barry, Feb 19 2003

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

Crossrefs

T(n,3) in triangle A080419.

Cf. A036068, A136158.

Programs

Magma

[(n+1)*(n+2)*(n+9)*3^n/18: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

Mathematica

Table[((n+1)(n+2)(n+9)3^n)/18,{n,0,30}] (* or *) LinearRecurrence[ {12,-54,108,-81},{1,10,66,360},30] (* Harvey P. Dale, Mar 21 2012 *)
CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

SageMath

[(n+1)*(n+2)*(n+9)*3^(n-2)/2 for n in range(31)] # G. C. Greubel, Dec 22 2023

Formula

G.f.: (1-2*x)/(1-3*x)^4.

a(n) = A006503(n+1)*3^(n-1).

a(n) = 12*a(n-1)-54*a(n-2)+108*a(n-3)-81*a(n-4). - Harvey P. Dale, Mar 21 2012

From G. C. Greubel, Dec 22 2023: (Start)

a(n) = (n+9)*A036068(n-1).

a(n) = A136158(n+3, 3).

E.g.f.: (1/2)*(2 + 14*x + 15*x^2 + 3*x^3)*exp(3*x). (End)

From Amiram Eldar, Jan 11 2024: (Start)

Sum_{n>=0} 1/a(n) = 44172*log(3/2)/7 - 20050659/7840.

Sum_{n>=0} (-1)^n/a(n) = 44496*log(4/3)/7 - 14329629/7840. (End)