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%I #58 Jun 12 2024 01:11:47
%S 1,-1,3,-1,27,119,1203,11759,136587,1771559,25562403,405657119,
%T 7022893947,131714582999,2660335750803,57570797728079,
%U 1328913670528107,32592691757218439,846383665814342403,23200396829831840639,669421949061096575067,20281206249626017421879
%N Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).
%H G. C. Greubel, <a href="/A344037/b344037.txt">Table of n, a(n) for n = 0..420</a>
%F a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A000670(k).
%F a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A008619(k).
%F a(n) = Sum_{k>=0} (k - 2)^n / 2^(k+1).
%F a(n) = (-2)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
%F a(n) ~ n! / (8 * log(2)^(n+1)). - _Vaclav Kotesovec_, Aug 15 2021
%t nmax = 21; CoefficientList[Series[Exp[-2 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[HurwitzLerchPhi[1/2, -n, -2]/2, {n, 0, 21}]
%t a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 40);
%o Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(x)) ))); // _G. C. Greubel_, Jun 11 2024
%o (SageMath)
%o def A344037_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
%o A344037_list(40) # _G. C. Greubel_, Jun 11 2024
%Y Cf. A000670, A007047, A008619, A052841, A330603, A346208.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Aug 01 2021
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