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A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).

%I A346740 #12 Jun 13 2024 01:42:31
%S A346740 1,-4,17,-75,340,-1573,7393,-35178,169035,-818603,3989250,-19538555,
%T A346740 96084397,-474052868,2344993157,-11624422855,57722000172,
%U A346740 -287012948441,1428705217949,-7118044107698,35489117143047,-177036294035559,883588566571138,-4411213326568599,22032317835916969
%N A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).
%H A346740 Seiichi Manyama, <a href="/A346740/b346740.txt">Table of n, a(n) for n = 0..581</a>
%F A346740 G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 5*x)).
%F A346740 a(n) = Sum_{k=0..n} binomial(n,k) * (-5)^(n-k) * Bell(k).
%F A346740 a(n) = exp(-1) * Sum_{k>=0} (k - 5)^n / k!.
%F A346740 a(0) = 1; a(n) = -5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).
%t A346740 nmax = 24; CoefficientList[Series[Exp[Exp[x] - 5 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A346740 Table[Sum[Binomial[n, k] (-5)^(n - k) BellB[k], {k, 0, n}], {n, 0, 24}]
%t A346740 a[0] = 1; a[n_] := a[n] = -5 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
%o A346740 (Magma)
%o A346740 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A346740 Coefficients(R!(Laplace( Exp(Exp(x) -5*x -1) ))) // _G. C. Greubel_, Jun 12 2024
%o A346740 (SageMath)
%o A346740 [factorial(n)*( exp(exp(x) -5*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jun 12 2024
%Y A346740 Cf. A000110, A000296, A126617, A196834, A196835, A290219, A346738, A346739.
%K A346740 sign
%O A346740 0,2
%A A346740 _Ilya Gutkovskiy_, Jul 31 2021