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A346208 Expansion of e.g.f.: exp(-3*x) / (2 - exp(x)).

%I A346208 #20 Jun 12 2024 01:11:39
%S A346208 1,-2,6,-14,54,-62,966,4786,71574,875938,12810726,202739986,
%T A346208 3511712694,65856494338,1330170266886,28785391689586,664456856787414,
%U A346208 16296345814039138,423191833100881446,11600198414334789586,334710974532291679734,10140603124807778534338
%N A346208 Expansion of e.g.f.: exp(-3*x) / (2 - exp(x)).
%H A346208 G. C. Greubel, <a href="/A346208/b346208.txt">Table of n, a(n) for n = 0..420</a>
%F A346208 a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000670(k).
%F A346208 a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A002620(k+2).
%F A346208 a(n) = Sum_{k>=0} (k - 3)^n / 2^(k+1).
%F A346208 a(n) = (-3)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
%F A346208 a(n) ~ n! / (16 * log(2)^(n+1)). - _Vaclav Kotesovec_, Aug 15 2021
%t A346208 nmax = 21; CoefficientList[Series[Exp[-3 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A346208 Table[HurwitzLerchPhi[1/2, -n, -3]/2, {n, 0, 21}]
%t A346208 a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
%o A346208 (Magma)
%o A346208 R<x>:=PowerSeriesRing(Rationals(), 40);
%o A346208 Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(x)) ))); // _G. C. Greubel_, Jun 11 2024
%o A346208 (SageMath)
%o A346208 def A346208_list(prec):
%o A346208     P.<x> = PowerSeriesRing(QQ, prec)
%o A346208     return P( exp(-3*x)/(2-exp(x)) ).egf_to_ogf().list()
%o A346208 A346208_list(40) # _G. C. Greubel_, Jun 11 2024
%Y A346208 Cf. A000670, A002620, A052841, A259533, A330603, A344037.
%K A346208 sign
%O A346208 0,2
%A A346208 _Ilya Gutkovskiy_, Aug 01 2021