%I #9 Jul 08 2017 00:45:48
%S 0,1,2,6,26,120,474,-3500,-169744,-4739628,-122528220,-3244006128,
%T -89971866744,-2643601630488,-82449886989120,-2730313541889120,
%U -95853665484598656,-3561107748108889344,-139703010646898138688,-5774800668716738596896,-250987866830927324395200
%N E.g.f. expansion of f(x) around x = 1, where f(x) is the coefficient from the tetration asymptotic: x^^n = x^^inf - f(x)*log(x^^inf)^n + O(log(x^^inf)^(2*n)).
%C The tetration x^^n is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). For x in [e^(-e), e^(1/e)] there is a limit x^^inf = limit_{n->inf} x^^n = e^(-W(-log x)), where W(z) is the Lambert W-function. The tetration approaches this limit exponentially: x^^n = x^^inf - f(x)*log(x^^inf)^n + O(log(x^^inf)^(2*n)), where the coefficient f(x) = lim_{n->inf} (x^^inf - x^^n)/log(x^^inf)^n depends on x. This sequence gives the e.g.f. expansion of f(x) around x = 1.
%H MathOverflow, <a href="http://mathoverflow.net/questions/260379/a-curious-series-related-to-the-asymptotic-behavior-of-the-tetration">Discussion of a related sequence</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%e f(x) = (1/1!)*(x-1) + (2/2!)*(x-1)^2 + (6/3!)*(x-1)^3 + (26/4!)*(x-1)^4 + (120/5!)*(x-1)^5 + ...
%t a[n_] := n! SeriesCoefficient[(Exp[-ProductLog[-Log[x]]] - Power @@ Table[x, {n}])/(-ProductLog[-Log[x]])^n, {x, 1, n}]; Table[a[n], {n, 0, 20}]
%Y Cf. A198094, A260691, A277435, A288607.
%K sign
%O 0,3
%A _Vladimir Reshetnikov_, Jun 11 2017