A080421 - OEIS

%I #25 Jan 11 2024 01:39:06

%S 1,10,66,360,1755,7938,34020,139968,557685,2165130,8227494,30705480,

%T 112842639,409209570,1466777160,5203870272,18294856425,63795240522,

%U 220829678730,759344158440,2595329855811,8821564534530,29832927334956,100419390748800,336561864306525

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

%H Vincenzo Librandi, <a href="/A080421/b080421.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-54,108,-81).

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

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

%F a(n) = 12*a(n-1)-54*a(n-2)+108*a(n-3)-81*a(n-4). - _Harvey P. Dale_, Mar 21 2012

%F From _G. C. Greubel_, Dec 22 2023: (Start)

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

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

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

%F From _Amiram Eldar_, Jan 11 2024: (Start)

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

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

%t 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 *)

%t CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^4, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 05 2013 *)

%o (Magma) [(n+1)*(n+2)*(n+9)*3^n/18: n in [0..30]]; // _Vincenzo Librandi_, Aug 05 2013

%o (SageMath) [(n+1)*(n+2)*(n+9)*3^(n-2)/2 for n in range(31)] # _G. C. Greubel_, Dec 22 2023

%Y T(n,3) in triangle A080419.

%Y Cf. A036068, A136158.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 19 2003