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%I #46 Jan 17 2020 17:44:53
%S 0,1,-3,11,-58,409,-3606,38149,-470856,6641793,-105398650,1858413061,
%T -36044759796,762659322385,-17481598316742,431535346662645,
%U -11413394655983536,321989729198400385,-9651573930139850610
%N E.g.f. log(1 + x*exp(-x)).
%C abs(a(n)) is the number of connected functions f:{1,2,...,n}->{1,2,...,n} such that every element is mapped into a recurrent element. Cf. A006153. - _Geoffrey Critzer_, May 24 2012
%H G. C. Greubel, <a href="/A009444/b009444.txt">Table of n, a(n) for n = 0..400</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=125">Encyclopedia of Combinatorial Structures 125</a>
%F abs(a(n)) is asymptotic to (n-1)!/LambertW(1)^n. - _Vladeta Jovovic_, Jul 12 2007
%F Sequence of absolute values has e.g.f. log(1/(1-x*exp(x))). - _Joerg Arndt_, Apr 30 2011
%F a(n) = (-1)^(n+1)*n!*sum(m=1..n, m^(n-m-1)/(n-m)!). - _Vladimir Kruchinin_, Oct 08 2011
%F a(n) = (-1)^(n + 1) * n + Sum_{k=1..n-1} (-1)^(n - k) * binomial(n-1,k-1) * (n - k) * a(k). - _Ilya Gutkovskiy_, Jan 17 2020
%t With[{nmax = 40}, CoefficientList[Series[Log[1 + x*Exp[-x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 22 2017 *)
%o (PARI) x='x+O('x^66); /* that many terms */
%o egf=1/(1+x/exp(x)); /* = 1 - x + 2*x^2 - 7/2*x^3 + 37/6*x^4 - 87/8*x^5 +... */
%o Vec(serlaplace(egf)) /* show terms */ /* _Joerg Arndt_, Apr 30 2011 */
%o (Maxima)
%o a(n):=(-1)^(n+1)*n!*sum(m^(n-m-1)/(n-m)!,m,1,n); /* _Vladimir Kruchinin_, Oct 08 2011 */
%o (Sage)
%o A009444 = lambda n: (-1)^(n+1)*factorial(n)*sum(m^(n-m-1)/factorial(n-m) for m in (1..n))
%o [A009444(n) for n in (0..9)] # _Peter Luschny_, Jan 18 2016
%Y Cf. A006153, A009306, A305990.
%K sign,easy
%O 0,3
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
%E Definition corrected by _Joerg Arndt_, Apr 30 2011
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