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A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).

%I A346738 #12 Jun 13 2024 01:48:18
%S A346738 1,-2,5,-13,36,-101,293,-848,2523,-7365,22402,-64395,205285,-541802,
%T A346738 2057617,-3403993,28685420,43885023,824532745,4878097904,44263112047,
%U A346738 357891860463,3169228222338,28506399763969,266822555964441,2573194635922990,25606751525353741
%N A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).
%H A346738 Seiichi Manyama, <a href="/A346738/b346738.txt">Table of n, a(n) for n = 0..579</a>
%F A346738 G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 3*x)).
%F A346738 a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * Bell(k).
%F A346738 a(n) = exp(-1) * Sum_{k>=0} (k - 3)^n / k!.
%F A346738 a(0) = 1; a(n) = -3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).
%t A346738 nmax = 26; CoefficientList[Series[Exp[Exp[x] - 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A346738 Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k], {k, 0, n}], {n, 0, 26}]
%t A346738 a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]
%o A346738 (Magma)
%o A346738 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A346738 Coefficients(R!(Laplace( Exp(Exp(x)-3*x-1) ))) // _G. C. Greubel_, Jun 12 2024
%o A346738 (SageMath)
%o A346738 [factorial(n)*( exp(exp(x)-3*x-1) ).series(x, n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jun 12 2024
%Y A346738 Cf. A000110, A000296, A005494, A126617, A193683, A290219, A346739, A346740.
%K A346738 sign
%O A346738 0,2
%A A346738 _Ilya Gutkovskiy_, Jul 31 2021