%I #23 Jul 24 2024 09:47:26
%S 1,18,129,646,2685,10002,34777,115566,372453,1175290,3654369,11245110,
%T 34349005,104373282,315969705,954002878,2874983541,8652474378,
%U 26015617585,78169534470,234766551261,704840716978,2115654610809,6349329417486,19052920751365,57169029907482
%N One fourth of third column (m=2) of triangle A142963.
%H Vincenzo Librandi, <a href="/A142965/b142965.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (10, -40, 82, -91, 52, -12).
%F a(n) = A142963(n+3,2)/4.
%F From _Johannes W. Meijer_, Feb 20 2009: (Start)
%F a(n) = 10a(n-1) - 40a(n-2) + 82a(n-3) - 91a(n-4) + 52a(n-5) - 12a(n-6).
%F a(n) = 35/2 + 2*n^2 + 12*n - 84*2^n - 24*2^n*n + 135/2*3^n
%F G.f.: (1 + 8*z - 11*z^2 - 6*z^3)/((1-z)^3*(1-2*z)^2*(1-3*z)).
%F (End)
%t LinearRecurrence[{10,-40,82,-91,52,-12}, {1,18,129,646,2685,10002},30] (* or *) CoefficientList[Series[(1+8x-11x^2-6x^3)/((x-1)^3 (2x-1)^2 (3x-1)),{x,0,30}],x] (* _Harvey P. Dale_, Apr 24 2011 *)
%o (Magma) [35/2+2*n^2+12*n-84*2^n-24*2^n*n+135/2*3^n: n in [0..25]]; // _Vincenzo Librandi_, Jun 18 2017
%Y Column m=1: 2*A142964; m=3: 8*A142966.
%Y From _Johannes W. Meijer_, Feb 20 2009: (Start)
%Y Cf. A156925.
%Y Equals A156920(n+2,2).
%Y Equals A156919(n+2,2)/2^n.
%Y (End)
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 15 2008