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%I #34 Mar 08 2024 12:00:21
%S 1,-1,3,-13,73,-501,4051,-37633,394353,-4596553,58941091,-824073141,
%T 12470162233,-202976401213,3535017524403,-65573803186921,
%U 1290434218669921,-26846616451246353,588633468315403843,-13564373693588558173,327697927886085654441
%N Expansion of e.g.f.: exp(-x/(1+x)).
%C For n >= 1, gives row sums of A008297, triangle of Lah numbers. - _Daniel Forgues_, Oct 12 2019
%H Seiichi Manyama, <a href="/A293125/b293125.txt">Table of n, a(n) for n = 0..444</a>
%H Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%F a(n) = (-1)^n * A000262(n).
%F From _Vaclav Kotesovec_, Sep 30 2017: (Start)
%F a(n) = -(2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2).
%F a(n) ~ (-1)^n * n^(n-1/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 5/(48*sqrt(n)) - 95/(4608*n)).
%F (End)
%F a(n) = (-1)^n *n! * Sum_{j=0..n-1} binomial(n-1, j)/(j+1)!, for n > 0. - _G. C. Greubel_, Dec 04 2018
%F a(n) = (-1)^n*n!*hypergeom([1 - n], [2], -1) for n > 0. - _Peter Luschny_, Oct 13 2019
%t CoefficientList[Series[E^(-x/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2017 *)
%t a[n_] := If[n == 0, 1, (-1)^n n! Hypergeometric1F1[1 - n, 2, -1]];
%t Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Oct 13 2019 )
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(-x/(1+x))))
%o (Magma) [1] cat [(-1)^n*Factorial(n)*(&+[Binomial(n-1, j)/Factorial(j+1): j in [0..n-1]]): n in [1..30]]; // _G. C. Greubel_, Dec 04 2018
%o (Sage) [1] + [(-1)^n*factorial(n)*sum(binomial(n-1,j)/factorial(j+1) for j in (0..n-1)) for n in (1..30)] # _G. C. Greubel_, Dec 04 2018
%o (GAP) a:=[-1,3];; for n in [3..25] do a[n]:=-(2*n-1)*a[n-1]-(n-2)*(n-1)*a[n-2]; od; Concatenation([1], a); # _G. C. Greubel_, Dec 04 2018
%Y Column k=0 of A293134.
%Y Cf. A000262, A008297.
%K sign
%O 0,3
%A _Seiichi Manyama_, Sep 30 2017
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