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A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).

%I A367977 #11 Jun 11 2024 19:13:19
%S A367977 1,1,9,73,849,12241,211929,4280473,98806689,2565862561,74035143849,
%T A367977 2349822967273,81361870604529,3051889548205681,123282485663042169,
%U A367977 5335770920836028473,246332487897909570369,12083010395805261921601,627555570373369525058889,34404109751876393769480073
%N A367977 Expansion of e.g.f. exp(-x) / (2 - exp(2*x)).
%H A367977 G. C. Greubel, <a href="/A367977/b367977.txt">Table of n, a(n) for n = 0..380</a>
%F A367977 a(n) = Sum_{k>=0} (2*k-1)^n / 2^(k+1).
%F A367977 a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 2^k * a(n-k).
%F A367977 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000670(k).
%t A367977 nmax = 19; CoefficientList[Series[Exp[-x]/(2 - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A367977 a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%o A367977 (Magma)
%o A367977 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A367977 Coefficients(R!(Laplace( Exp(-x)/(2-Exp(2*x)) ))) // _G. C. Greubel_, Jun 10 2024
%o A367977 (SageMath)
%o A367977 def A367977_list(prec):
%o A367977     P.<x> = PowerSeriesRing(QQ, prec)
%o A367977     return P( exp(-x)/(2-exp(2*x)) ).egf_to_ogf().list()
%o A367977 A367977_list(50) # _G. C. Greubel_, Jun 10 2024
%Y A367977 Cf. A000670, A052841, A080253, A216794, A344037, A367979, A367980, A367981, A367982, A367983.
%K A367977 nonn
%O A367977 0,3
%A A367977 _Ilya Gutkovskiy_, Dec 07 2023