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A367981 Expansion of e.g.f. exp(-x) / (2 - exp(4*x)).

%I A367981 #10 Jun 11 2024 19:19:44
%S A367981 1,3,41,699,16145,465843,16129721,651567339,30080413985,1562287110243,
%T A367981 90156154697801,5722984840599579,396313048731199025,
%U A367981 29731461242293159443,2402032174787943537881,207923994397338180409419,19198138598327305954291265,1883400035807194281432757443
%N A367981 Expansion of e.g.f. exp(-x) / (2 - exp(4*x)).
%H A367981 G. C. Greubel, <a href="/A367981/b367981.txt">Table of n, a(n) for n = 0..345</a>
%F A367981 a(n) = Sum_{k>=0} (4*k-1)^n / 2^(k+1).
%F A367981 a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
%F A367981 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 4^k * A000670(k).
%t A367981 nmax = 17; CoefficientList[Series[Exp[-x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A367981 a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
%o A367981 (Magma)
%o A367981 R<x>:=PowerSeriesRing(Rationals(), 40);
%o A367981 Coefficients(R!(Laplace( Exp(-x)/(2-Exp(4*x)) ))); // _G. C. Greubel_, Jun 11 2024
%o A367981 (SageMath)
%o A367981 def A367981_list(prec):
%o A367981     P.<x> = PowerSeriesRing(QQ, prec)
%o A367981     return P( exp(-x)/(2-exp(4*x)) ).egf_to_ogf().list()
%o A367981 A367981_list(40) # _G. C. Greubel_, Jun 11 2024
%Y A367981 Cf. A000670, A052841, A285067, A328183, A367977, A367979, A367980, A367982, A367983.
%K A367981 nonn
%O A367981 0,2
%A A367981 _Ilya Gutkovskiy_, Dec 07 2023