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A080421
a(n) = (n+1)*(n+2)*(n+9)*3^n/18.
7
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
0,2
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)
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 *)
PROG
(Magma) [(n+1)*(n+2)*(n+9)*3^n/18: n in [0..30]]; // 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
CROSSREFS
T(n,3) in triangle A080419.
Sequence in context: A269468 A024391 A074362 * A320817 A004310 A026853
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 19 2003
STATUS
approved