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A277523
Decimal expansion of the second derivative of the infinite power tower function x^x^x... at x = 1/2.
10
2, 2, 2, 1, 4, 0, 2, 1, 3, 6, 0, 1, 2, 2, 2, 1, 2, 6, 5, 5, 1, 5, 5, 3, 7, 3, 8, 5, 9, 6, 8, 0, 0, 3, 0, 8, 9, 5, 9, 1, 0, 8, 9, 7, 2, 6, 8, 6, 2, 8, 1, 5, 1, 7, 3, 8, 4, 7, 4, 4, 7, 7, 9, 8, 7, 0, 2, 1, 3, 9, 6, 9, 1, 7, 4, 7, 8, 5, 5, 1, 9, 0, 3, 9, 7, 5, 7, 2, 6, 5, 4, 2, 4, 2, 7, 1, 7, 8, 8, 4, 5, 2, 2, 5, 4
OFFSET
0,1
LINKS
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.
FORMULA
Equals 4 * LambertW(log(2))^2 * ((2-log(2)) * LambertW(log(2))^2 + (3-2*log(2)) * LambertW(log(2))-log(2)) / (log(2) * (1+LambertW(log(2))))^3. - Vladimir Reshetnikov, Oct 20 2016
EXAMPLE
0.222140213601222126551553738596800308959108972686281...
MATHEMATICA
RealDigits[4 LambertW[Log[2]]^2 ((2 - Log[2]) LambertW[Log[2]]^2 + (3 - 2 Log[2]) LambertW[Log[2]] - Log[2])/(Log[2] (1 + LambertW[Log[2]]))^3, 10, 105][[1]] (* Vladimir Reshetnikov, Oct 20 2016 *)
f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f''[1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)
PROG
(PARI) 4*lambertw(log(2))^2*((2-log(2))*lambertw(log(2))^2 + (3-2*log(2)) *lambertw(log(2))-log(2))/(log(2)*(1+lambertw(log(2))))^3 \\ G. C. Greubel, Nov 10 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Alois P. Heinz, Oct 19 2016
STATUS
approved