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%I #18 Feb 07 2024 01:33:47
%S 5,9,0,6,1,6,1,0,9,1,4,9,6,4,1,2,4,9,7,4,3,8,0,6,9,0,9,3,2,3,2,5,1,5,
%T 5,7,1,1,6,6,5,3,0,4,8,8,7,3,8,8,0,0,6,7,4,4,0,2,7,9,2,0,1,9,2,1,8,2,
%U 4,9,3,3,7,5,4,4,5,7,2,7,5,2,5,4,4,3,5,2,2,3,9,4,1,8,4,8,8,3,8,6,2,6,8,9
%N Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/4.
%C It is known that (1/4)^(1/4)^(1/4)^... = 1/2.
%H Alois P. Heinz, <a href="/A277651/b277651.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 1/(1 + log(2)) = 1/(1 + A002162).
%F Equals Integral_{x >= 0} exp(-x)/2^x dx. - _Peter Bala_, Feb 05 2024
%e 0.59061610914964124974380690932325155711665304887388...
%t RealDigits[1/(1 + Log[2]), 10, 120][[1]]
%t (* or *)
%t f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f'[1/4], 10, 120][[1]] (* _Amiram Eldar_, May 23 2023 *)
%Y Cf. A002162, A277522, A277559, A293009, A300916.
%K nonn,cons
%O 0,1
%A _Alois P. Heinz_, Oct 25 2016
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