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%I #12 Jul 23 2021 02:36:06
%S 3,25,130,546,2037,7071,23436,75328,237127,735813,2260518,6896046,
%T 20933673,63325051,191088976,575625900,1731858075,5206059585,
%U 15640198410,46966732090,140996664733,423191320215,1269993390420
%N Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.
%H G. C. Greubel, <a href="/A112495/b112495.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) = 3*a(n-1)+ (n+3)*(2^(n+2)-(n+3)), n>=1, a(0)=3.
%F G.f.: (3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x)).
%F a(n) = 3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2. - _Vaclav Kotesovec_, Jul 23 2021
%t CoefficientList[Series[(3 - 5*x)/(((1 - x)^3)*((1 - 2*x)^2)*(1 - 3*x)), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 13 2017 *)
%t Table[3^(n+4)/2 - (n+6)*2^(n+3) + n^2/2 + 9*n/2 + 21/2, {n,0,25}] (* _Vaclav Kotesovec_, Jul 23 2021 *)
%o (PARI) x='x+O('x^50); Vec((3-5*x)/(((1-x)^3)*((1-2*x)^2)*(1-3*x))) \\ _G. C. Greubel_, Nov 13 2017
%Y Cf. A000295 (second column).
%Y Column k=2 of A124324 (shifted).
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Oct 14 2005
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