%I #38 Sep 08 2022 08:45:09

%S 1,3,12,40,120,336,896,2304,5760,14080,33792,79872,186368,430080,

%T 983040,2228224,5013504,11206656,24903680,55050240,121110528,

%U 265289728,578813952,1258291200,2726297600,5888802816,12683575296,27246198784

%N Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).

%C The third column of number triangle A080928.

%H Vincenzo Librandi, <a href="/A080929/b080929.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 (6,-12,8).

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

%F For n>0, a(n) = (n+1)*(n+2)*2^(n-2). - _Ralf Stephan_, Jan 16 2004

%F a(n) = Sum_{k=0..n} Sum_{i=0..n} (k+1)*binomial(n-1,i). - _Wesley Ivan Hurt_, Sep 20 2017

%F From _Amiram Eldar_, Jan 07 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 7 - 8*log(2).

%F Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)

%p [seq (ceil(binomial(n+2,2)*2^(n-1)),n=0..30)]; # _Zerinvary Lajos_, Nov 01 2006

%t CoefficientList[Series[(1-x)(1-2x+4x^2)/(1-2x)^3, {x,0,30}], x] (* _Michael De Vlieger_, Sep 21 2017 *)

%t Join[{1}, LinearRecurrence[{6,-12,8}, {3,12,40}, 30]] (* _G. C. Greubel_, Jul 23 2019 *)

%o (Magma) [n eq 0 select 1 else (n+1)*(n+2)*2^(n-2): n in [0..30]]; // _Vincenzo Librandi_, Sep 22 2011

%o (PARI) vector(30, n, n--; if(n==0,1, 2^(n-1)*binomial(n+2,2) )) \\ _G. C. Greubel_, Jul 23 2019

%o (Sage) [1]+[2^(n-1)*binomial(n+2,2) for n in (1..30)] # _G. C. Greubel_, Jul 23 2019

%o (GAP) Concatenation([1], List([1..30], n-> 2^(n-1)*Binomial(n+2,2))); # _G. C. Greubel_, Jul 23 2019

%Y Essentially the same as A052482.

%Y Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Feb 26 2003