A346208 - OEIS

%I #20 Jun 12 2024 01:11:39

%S 1,-2,6,-14,54,-62,966,4786,71574,875938,12810726,202739986,

%T 3511712694,65856494338,1330170266886,28785391689586,664456856787414,

%U 16296345814039138,423191833100881446,11600198414334789586,334710974532291679734,10140603124807778534338

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

%H G. C. Greubel, <a href="/A346208/b346208.txt">Table of n, a(n) for n = 0..420</a>

%F a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000670(k).

%F a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A002620(k+2).

%F a(n) = Sum_{k>=0} (k - 3)^n / 2^(k+1).

%F a(n) = (-3)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).

%F a(n) ~ n! / (16 * log(2)^(n+1)). - _Vaclav Kotesovec_, Aug 15 2021

%t nmax = 21; CoefficientList[Series[Exp[-3 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[HurwitzLerchPhi[1/2, -n, -3]/2, {n, 0, 21}]

%t a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 40);

%o Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(x)) ))); // _G. C. Greubel_, Jun 11 2024

%o (SageMath)

%o def A346208_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( exp(-3*x)/(2-exp(x)) ).egf_to_ogf().list()

%o A346208_list(40) # _G. C. Greubel_, Jun 11 2024

%Y Cf. A000670, A002620, A052841, A259533, A330603, A344037.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Aug 01 2021