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A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).

%I A346739 #13 Jun 13 2024 01:42:25
%S A346739 1,-3,10,-35,127,-472,1787,-6855,26572,-103765,407695,-1608378,
%T A346739 6369117,-25271183,100542930,-400114103,1597052419,-6359524256,
%U A346739 25481982047,-101103395443,409291679676,-1592903606657,6729506287091,-23748796926026,123501587468073,-227183793907851
%N A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).
%H A346739 Seiichi Manyama, <a href="/A346739/b346739.txt">Table of n, a(n) for n = 0..580</a>
%F A346739 G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 4*x)).
%F A346739 a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * Bell(k).
%F A346739 a(n) = exp(-1) * Sum_{k>=0} (k - 4)^n / k!.
%F A346739 a(0) = 1; a(n) = -4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).
%t A346739 nmax = 25; CoefficientList[Series[Exp[Exp[x] - 4 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A346739 Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k], {k, 0, n}], {n, 0, 25}]
%t A346739 a[0] = 1; a[n_] := a[n] = -4 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]
%o A346739 (Magma)
%o A346739 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A346739 Coefficients(R!(Laplace( Exp(Exp(x) -4*x -1) ))) // _G. C. Greubel_, Jun 12 2024
%o A346739 (SageMath)
%o A346739 [factorial(n)*( exp(exp(x) -4*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jun 12 2024
%Y A346739 Cf. A000110, A000296, A045379, A126617, A193684, A290219, A346738, A346740.
%K A346739 sign
%O A346739 0,2
%A A346739 _Ilya Gutkovskiy_, Jul 31 2021