A128961 - OEIS

%I #25 Oct 02 2022 10:32:44

%S 0,54,648,4860,29160,153090,734832,3306744,14171760,58458510,

%T 233834040,911952756,3482001432,13057505370,48212327520,175630621680,

%U 632270238048,2252462723046,7949868434280,27824539519980,96653663595720,333455139405234,1143274763675088

%N a(n) = (n^3 - n)*3^n.

%H Vincenzo Librandi, <a href="/A128961/b128961.txt">Table of n, a(n) for n = 1..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.: 54*x^2/(1-3*x)^4. - _Vincenzo Librandi_, Feb 12 2013

%F a(n) = 12*a(n-1) - 54*a(n-2) + 108*a(n-3) - 81*a(n-4). - _Vincenzo Librandi_, Feb 12 2013

%F From _Amiram Eldar_, Oct 02 2022: (Start)

%F a(n) = A007531(n+1)*A000244(n).

%F Sum_{n>=2} 1/a(n) = (2/3)*log(3/2) - 1/4.

%F Sum_{n>=2} (-1)^n/a(n) = (8/3)*log(4/3) - 3/4. (End)

%t LinearRecurrence[{12, -54, 108, -81}, {0, 54, 648, 4860}, 30] (* or *) CoefficientList[Series[54 x/(1 - 3 x)^4, {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 12 2013 *)

%o (Magma) [(n^3-n)*3^n: n in [1..25]]; /* or */ I:=[0,54,648,4860]; [n le 4 select I[n] else 12*Self(n-1)-54*Self(n-2)+108*Self(n-3)-81*Self(n-4): n in [1..25]]; // _Vincenzo Librandi_, Feb 12 2013

%Y Cf. A000244, A007531, A036289, A128796.

%Y Cf. A128960, A128962, A128963, A128964, A128965, A128967, A128969.

%K nonn,easy

%O 1,2

%A _Mohammad K. Azarian_, Apr 28 2007

%E Offset corrected by _Mohammad K. Azarian_, Nov 20 2008