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

%I A277523 #24 Feb 16 2025 08:33:37
%S A277523 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,
%T A277523 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,
%U A277523 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
%N A277523 Decimal expansion of the second derivative of the infinite power tower function x^x^x... at x = 1/2.
%H A277523 Alois P. Heinz, <a href="/A277523/b277523.txt">Table of n, a(n) for n = 0..1000</a>
%H A277523 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%H A277523 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F A277523 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
%e A277523 0.222140213601222126551553738596800308959108972686281...
%t A277523 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 *)
%t A277523 f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f''[1/2], 10, 120][[1]] (* _Amiram Eldar_, May 23 2023 *)
%o A277523 (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
%Y A277523 Cf. A033917, A104748 (0th derivative), A277522, A277524, A277525, A277526, A277527, A277528, A277529, A277530, A277531.
%K A277523 nonn,cons
%O A277523 0,1
%A A277523 _Alois P. Heinz_, Oct 19 2016