A293125 Expansion of e.g.f.: exp(-x/(1+x)).
1, -1, 3, -13, 73, -501, 4051, -37633, 394353, -4596553, 58941091, -824073141, 12470162233, -202976401213, 3535017524403, -65573803186921, 1290434218669921, -26846616451246353, 588633468315403843, -13564373693588558173, 327697927886085654441
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..444
- Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
Programs
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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
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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
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Mathematica
CoefficientList[Series[E^(-x/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *) a[n_] := If[n == 0, 1, (-1)^n n! Hypergeometric1F1[1 - n, 2, -1]]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Oct 13 2019 *) -
PARI
my(x='x+O('x^21)); Vec(serlaplace(exp(-x/(1+x)))) -
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
Formula
a(n) = (-1)^n * A000262(n).
From Vaclav Kotesovec, Sep 30 2017: (Start)
a(n) = -(2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2).
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)). (End)
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
a(n) = (-1)^n*n!*hypergeom([1 - n], [2], -1) for n > 0. - Peter Luschny, Oct 13 2019
Comments