login
Fourth column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.
4

%I #12 Jul 23 2021 02:46:19

%S 15,210,1750,11368,63805,325930,1561516,7150000,31682651,137031986,

%T 582035714,2438479592,10109790809,41579014154,169946747160,

%U 691299506640,2801567046135,11320801495410,45642930545070,183698923750440

%N Fourth column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.

%H G. C. Greubel, <a href="/A112496/b112496.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (20, -175, 882, -2835, 6072, -8777, 8458, -5204, 1848, -288).

%F G.f.: (15-90*x+175*x^2-112*x^3)/((1-x)^4*(1-2*x)^3*(1-3*x)^2*(1-4*x)).

%F a(n) = 4*a(n-1) + (n+5)*A112495(n).

%F a(n) = 2^(2*n+11)/3- 3^(n+5)*(n+9)/2 + 2^(n+3)*(n^2 + 15*n + 58) - n^3/6 - 3*n^2 - 55*n/3 - 229/6. - _Vaclav Kotesovec_, Jul 23 2021

%t CoefficientList[Series[(15 - 90*x + 175*x^2 - 112*x^3)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2*(1 - 4*x)), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 13 2017 *)

%t Table[2^(2*n+11)/3- 3^(n+5)*(n+9)/2 + 2^(n+3)*(n^2 + 15*n + 58) - n^3/6 - 3*n^2 - 55*n/3 - 229/6, {n,0,25}] (* _Vaclav Kotesovec_, Jul 23 2021 *)

%o (PARI) x='x+O('x^50); Vec((15-90*x+175*x^2-112*x^3)/((1-x)^4*(1-2*x)^3*(1-3*x)^2*(1-4*x))) \\ _G. C. Greubel_, Nov 13 2017

%Y Cf. A112495 (third column).

%Y Column k=3 of A124324 (shifted).

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Oct 14 2005