A277523 Decimal expansion of the second derivative of the infinite power tower function x^x^x... at x = 1/2.

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

Cf. A033917, A104748 (0th derivative), A277522, A277524, A277525, A277526, A277527, A277528, A277529, A277530, A277531.

Sequence in context: A127496 A350189 A289778 * A144393 A351112 A089400

Adjacent sequences: A277520 A277521 A277522 * A277524 A277525 A277526

Keyword

nonn,cons

Author

Alois P. Heinz, Oct 19 2016

Status

approved