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i hope you dont mind this is not about tetration.

let f(x) be a taylor series with radius 1 and f(0) = 1
f(x) is strictly positive real in [-1,1] and nondecreasing in [0,1].

f(x) = 1 + a1 x + a2 x^2 + ...

f(x)^2 = 1 + b1 x + b2 x^2 + ...

f(x)^(1/2) = 1 + c1 x + c2 x^2 + ... ( the 1 means we take the positive root )

find a_n such that b_n * c_n = 1

i recall that srinivasa ramanujan did similar investigations so maybe i missed something trivial.

regards

tommy1729