A290219 - OEIS

%I #20 Jun 12 2024 21:05:40

%S 1,0,2,-13,127,-1573,23711,-421356,8626668,-199971255,5177291275,

%T -148078588667,4636966634653,-157786054331852,5797411243015250,

%U -228749440644895405,9646951350227609155,-433035586385769361001,20614401475233006857035,-1037331650810058231498688

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

%C The n-th term of the n-th inverse binomial transform of A000110.

%H Seiichi Manyama, <a href="/A290219/b290219.txt">Table of n, a(n) for n = 0..386</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) ~ (-1)^n * exp(exp(-1) - 1) * n^n. - _Vaclav Kotesovec_, Aug 04 2021

%p b:= proc(n, k) option remember; `if`(n=0, 1,

%p       k*b(n-1, k)+ b(n-1, k+1))

%p     end:

%p a:= n-> b(n, -n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2021

%t Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]

%t Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 50);

%o A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;

%o [A290219(n): n in [0..30]]; // _G. C. Greubel_, Jun 12 2024

%o (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 Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.

%Y Main diagonal of A361781.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Oct 06 2017