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A288606
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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)).
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1
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0, 1, 2, 6, 26, 120, 474, -3500, -169744, -4739628, -122528220, -3244006128, -89971866744, -2643601630488, -82449886989120, -2730313541889120, -95853665484598656, -3561107748108889344, -139703010646898138688, -5774800668716738596896, -250987866830927324395200
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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 + ...
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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