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A317096 Expansion of e.g.f. ((1 - x)/(1 - 2*x))*exp(x/(x - 1)).

%I A317096 #13 Mar 09 2021 20:14:57
%S A317096 1,0,1,8,69,704,8485,118824,1900297,34191296,683657001,15038537480,
%T A317096 360903291661,9383240195328,262727926084429,7881806223689384,
%U A317096 252217461390469905,8575390623429206144,308714050531090308817,11731134397549023854856,469245396934886909801941,19708307298664103361642560
%N A317096 Expansion of e.g.f. ((1 - x)/(1 - 2*x))*exp(x/(x - 1)).
%C A317096 Lah transform of A000166.
%H A317096 G. C. Greubel, <a href="/A317096/b317096.txt">Table of n, a(n) for n = 0..400</a>
%H A317096 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A317096 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A317096 a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000166(k)*n!/k!.
%F A317096 a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n+1). - _Vaclav Kotesovec_, Mar 26 2019
%F A317096 a(n) = n!*(LaguarreL(n,1) + Sum_{j=0..n-2} 2^(n-j-2)*LaguerreL(j,1)). - _G. C. Greubel_, Mar 09 2021
%p A317096 a:=series(exp(x/(x - 1))*(1 - x)/(1 - 2*x), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # _Paolo P. Lava_, Mar 26 2019
%t A317096 nmax = 21; CoefficientList[Series[Exp[x/(x - 1)] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
%t A317096 Table[Sum[Binomial[n - 1, k - 1] Subfactorial[k] n!/k!, {k, 0, n}], {n, 0, 21}]
%t A317096 A317096[n_]:= A317096[n]= n!*(LaguerreL[n,1] + Sum[2^(n-j-2)*LaguerreL[j,1], {j,0, n-2}]); Table[A317096[n], {n,0,25}] (* _G. C. Greubel_, Mar 09 2021 *)
%o A317096 (Sage)
%o A317096 def A317096(n): return factorial(n)*( gen_laguerre(n,0,1) + sum(2^(n-j-2)*gen_laguerre(j,0,1) for j in (0..n-2)) )
%o A317096 [A317096(n) for n in (0..25)] # _G. C. Greubel_, Mar 09 2021
%o A317096 (Magma)
%o A317096 R<x>:=PowerSeriesRing(Rationals(), 26);
%o A317096 Coefficients(R!(Laplace( ((1-x)/(1-2*x))*Exp(x/(x-1)) ))); // _G. C. Greubel_, Mar 09 2021
%Y A317096 Cf. A000166, A002866, A052852, A059115.
%K A317096 nonn
%O A317096 0,4
%A A317096 _Ilya Gutkovskiy_, Aug 01 2018