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%I #19 Oct 03 2023 10:31:45
%S 1,3,11,58,409,3606,38149,470856,6641793,105398650,1858413061,
%T 36044759796,762659322385,17481598316742,431535346662645,
%U 11413394655983536,321989729198400385,9651573930139850610,306321759739045148293,10262156907184058219340
%N Expansion of e.g.f.: (1+x) / (exp(-x) - x).
%F a(n) ~ n! / LambertW(1)^(n+1).
%F a(n) = (-1)^n * A009444(n+1).
%F a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - _Detlef Meya_, Sep 05 2023
%t nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
%t a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* _Detlef Meya_, Sep 05 2023 *)
%Y Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Jun 16 2018
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