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%I #28 May 23 2023 05:39:33
%S 5,6,9,2,4,5,2,0,4,4,2,6,3,4,8,0,6,1,0,6,5,3,3,0,4,7,7,8,4,1,9,6,6,9,
%T 0,5,2,6,3,8,6,5,9,7,3,1,4,6,3,0,2,7,6,4,5,4,9,8,0,1,1,6,7,9,1,8,3,1,
%U 1,5,4,2,1,1,3,4,6,7,0,8,7,6,2,3,2,4,0,6,7,4,8,7,9,2,6,3,5,0,2,4,5,1,5,0,3
%N Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/2.
%H Alois P. Heinz, <a href="/A277522/b277522.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%F Equals 2*exp(-2*LambertW(log(2)))/(1+LambertW(log(2))). - _Vladimir Reshetnikov_, Oct 20 2016
%e 0.5692452044263480610653304778419669052638659731463...
%t RealDigits[2 Exp[-2 ProductLog[Log[2]]]/(1 + ProductLog[Log[2]]), 10, 105][[1]] (* _Vladimir Reshetnikov_, Oct 20 2016 *)
%t f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f'[1/2], 10, 120][[1]] (* _Amiram Eldar_, May 23 2023 *)
%o (PARI) 2*exp(-2*lambertw(log(2)))/(1+lambertw(log(2))) \\ _G. C. Greubel_, Nov 10 2017
%Y Cf. A033917, A104748, A277523, A277524, A277525, A277526, A277527, A277528, A277529, A277530, A277531, A277559, A277651, A293009, A300916.
%K nonn,cons
%O 0,1
%A _Alois P. Heinz_, Oct 19 2016
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