A288606 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)).

0, 1, 2, 6, 26, 120, 474, -3500, -169744, -4739628, -122528220, -3244006128, -89971866744, -2643601630488, -82449886989120, -2730313541889120, -95853665484598656, -3561107748108889344, -139703010646898138688, -5774800668716738596896, -250987866830927324395200

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..20.

MathOverflow, Discussion of a related sequence.

Eric Weisstein's World of Mathematics, Power Tower.

Wikipedia, Lambert W function.

Wikipedia, Tetration.

Example

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 + ...

Mathematica

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}]

Crossrefs

Cf. A198094, A260691, A277435, A288607.

Sequence in context: A114710 A230245 A358107 * A379163 A377501 A208034

Adjacent sequences: A288603 A288604 A288605 * A288607 A288608 A288609

Keyword

sign

Author

Vladimir Reshetnikov, Jun 11 2017

Status

approved