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a(n) = 3^n * n*(n + 1).
2

%I #30 Nov 22 2024 16:26:05

%S 0,6,54,324,1620,7290,30618,122472,472392,1771470,6495390,23383404,

%T 82904796,290166786,1004423490,3443737680,11708708112,39516889878,

%U 132497807238,441659357460,1464449448420,4832683179786,15878816162154

%N a(n) = 3^n * n*(n + 1).

%H Vincenzo Librandi, <a href="/A116138/b116138.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).

%F From _R. J. Mathar_, Dec 19 2008: (Start)

%F G.f.: 6*x/(1-3*x)^3.

%F a(n) = 6 * A027472(n+2). (End)

%F a(n) = 9*a(n-1) -27*a(n-2) +27*a(n-3). - _Vincenzo Librandi_, Feb 28 2013

%F E.g.f.: 3*x*(2 + 3*x)*exp(3*x). - _G. C. Greubel_, May 10 2019

%F From _Amiram Eldar_, Jul 20 2020: (Start)

%F Sum_{n>=1} 1/a(n) = 1 - 2*log(3/2).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(4/3) - 1. (End)

%t Table[(n^2 + n) 3^n, {n, 0, 30}] (* _Vincenzo Librandi_, Feb 28 2013 *)

%o (Magma) [(n^2+n)*3^n: n in [0..30]]; // _Vincenzo Librandi_, Feb 28 2013

%o (Magma) I:=[0,6,54]; [n le 3 select I[n] else 9*Self(n-1)-27*Self(n-2)+27*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Feb 28 2013

%o (PARI) a(n)=(n^2+n)*3^n \\ _Charles R Greathouse IV_, Feb 28 2013

%o (Sage) [3^n*n*(n+1) for n in (0..30)] # _G. C. Greubel_, May 10 2019

%o (GAP) List([0..30], n-> 3^n*n*(n+1)); # _G. C. Greubel_, May 10 2019

%Y Cf. A007758, A036289, A128796, A027472.

%K nonn,easy

%O 0,2

%A _Mohammad K. Azarian_, Apr 08 2007