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%I #10 Jun 11 2024 19:10:47
%S 1,1,19,217,3835,82801,2150659,65156617,2256029515,87878584801,
%T 3803459964499,181078683329017,9404687464288795,529155742667806801,
%U 32063235363798322339,2081586179439325213417,144148514796485770141675,10606079719868369436964801,826272285216863547170504179
%N Expansion of e.g.f. exp(-2*x) / (2 - exp(3*x)).
%H G. C. Greubel, <a href="/A367980/b367980.txt">Table of n, a(n) for n = 0..360</a>
%F a(n) = Sum_{k>=0} (3*k-2)^n / 2^(k+1).
%F a(n) = (-2)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
%F a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * 3^k * A000670(k).
%t nmax = 18; CoefficientList[Series[Exp[-2 x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 40);
%o Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(3*x)) ))); // _G. C. Greubel_, Jun 11 2024
%o (SageMath)
%o def A367980_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( exp(-2*x)/(2-exp(3*x)) ).egf_to_ogf().list()
%o A367980_list(40) # _G. C. Greubel_, Jun 11 2024
%Y Cf. A000670, A328182, A344037, A355218, A367977, A367979, A367981, A367982, A367983.
%K nonn,changed
%O 0,3
%A _Ilya Gutkovskiy_, Dec 07 2023
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