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A091540 Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).

%I A091540 #13 Sep 08 2022 08:45:13
%S A091540 1,13,184,3040,58360,1283800,31917760,886123840,27192323200,
%T A091540 914387689600,33446228569600,1322364153510400,56203860301388800,
%U A091540 2555756347720576000,123819357959385088000,6367367706293321728000
%N A091540 Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).
%C A091540 A certain difference of two triple factorial sequences.
%C A091540 If offset 0: exponential (also called binomial) convolution of A091541 and A051606.
%H A091540 G. C. Greubel, <a href="/A091540/b091540.txt">Table of n, a(n) for n = 2..380</a>
%F A091540 a(n)= (5*2/fac3(3*n-1))*A091539(n), n>=2, with fac3(3*n-1) := A008544(n) (triple factorials).
%F A091540 E.g.f.: (1-2*x-(1-3*x)^(2/3))/(2*(1-3*x))= (1/2-x+int((1-3*x)^(-1/3), x))/(1-3*x).
%F A091540 E.g.f. with offset 0: (3-2*(1-3*x)^(2/3))/(1-3*x)^3.
%F A091540 a(n)=(fac3(3*n) - 3*fac3(3*n-2))/3! with fac3(3*n) := A032031(n)= n!*3^n and fac3(3*n-2) := A007559(n).
%F A091540 a(n) ~ 3^(n-1) * n! / 2. - _Vaclav Kotesovec_, Aug 16 2018
%t A091540 Drop[With[{nmax = 50}, CoefficientList[Series[(1 - 2*x - (1 - 3*x)^(2/3))/(2*(1 - 3*x)), {x, 0, nmax}], x]*Range[0, nmax]!],2] (* _G. C. Greubel_, Aug 15 2018 *)
%o A091540 (PARI) x='x+O('x^30); Vec(serlaplace((1 - 2*x - (1 - 3*x)^(2/3))/(2*(1 - 3*x)))) \\ _G. C. Greubel_, Aug 15 2018
%o A091540 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3-2*(1-3*x)^(2/3))/(1-3*x)^3 )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 15 2018
%Y A091540 Cf. A091541.
%K A091540 nonn,easy
%O A091540 2,2
%A A091540 _Wolfdieter Lang_, Feb 13 2004