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A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

%I A290219 #20 Jun 12 2024 21:05:40
%S A290219 1,0,2,-13,127,-1573,23711,-421356,8626668,-199971255,5177291275,
%T A290219 -148078588667,4636966634653,-157786054331852,5797411243015250,
%U A290219 -228749440644895405,9646951350227609155,-433035586385769361001,20614401475233006857035,-1037331650810058231498688
%N A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).
%C A290219 The n-th term of the n-th inverse binomial transform of A000110.
%H A290219 Seiichi Manyama, <a href="/A290219/b290219.txt">Table of n, a(n) for n = 0..386</a>
%H A290219 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A290219 a(n) ~ (-1)^n * exp(exp(-1) - 1) * n^n. - _Vaclav Kotesovec_, Aug 04 2021
%p A290219 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A290219       k*b(n-1, k)+ b(n-1, k+1))
%p A290219     end:
%p A290219 a:= n-> b(n, -n):
%p A290219 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2021
%t A290219 Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]
%t A290219 Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]
%o A290219 (Magma)
%o A290219 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A290219 A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;
%o A290219 [A290219(n): n in [0..30]]; // _G. C. Greubel_, Jun 12 2024
%o A290219 (SageMath) [factorial(n)*( exp(exp(x) -n*x -1) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jun 12 2024
%Y A290219 Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.
%Y A290219 Main diagonal of A361781.
%K A290219 sign
%O A290219 0,3
%A A290219 _Ilya Gutkovskiy_, Oct 06 2017