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A367979 Expansion of e.g.f. exp(-x) / (2 - exp(3*x)).

%I A367979 #11 Jun 12 2024 01:11:34
%S A367979 1,2,22,278,4822,104342,2709622,82092278,2842418902,110720079062,
%T A367979 4792059271222,228144844817078,11849163703935382,666694458859845782,
%U A367979 40397145162583154422,2622634244645856386678,181615748103175019442262,13362823095925278064444502,1041037845089466806646007222
%N A367979 Expansion of e.g.f. exp(-x) / (2 - exp(3*x)).
%H A367979 G. C. Greubel, <a href="/A367979/b367979.txt">Table of n, a(n) for n = 0..360</a>
%F A367979 a(n) = Sum_{k>=0} (3*k-1)^n / 2^(k+1).
%F A367979 a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
%F A367979 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * A000670(k).
%t A367979 nmax = 18; CoefficientList[Series[Exp[-x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A367979 a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%o A367979 (Magma)
%o A367979 R<x>:=PowerSeriesRing(Rationals(), 40);
%o A367979 Coefficients(R!(Laplace( Exp(-x)/(2-Exp(3*x)) ))); // _G. C. Greubel_, Jun 11 2024
%o A367979 (SageMath)
%o A367979 def A367979_list(prec):
%o A367979     P.<x> = PowerSeriesRing(QQ, prec)
%o A367979     return P( exp(-x)/(2-exp(3*x)) ).egf_to_ogf().list()
%o A367979 A367979_list(40) # _G. C. Greubel_, Jun 11 2024
%Y A367979 Cf. A000670, A052841, A151919, A328182, A346208, A367977, A367980, A367981, A367982, A367983.
%K A367979 nonn
%O A367979 0,2
%A A367979 _Ilya Gutkovskiy_, Dec 07 2023