OFFSET
0,1
COMMENTS
f(A199203) = 1, where f is the functional square root of exponent.
The listed digits are a conjecture obtained as a common result of several different numeric approximation algorithms, they haven't been rigorously proved to be correct.
LINKS
Citizendium, Superfunction.
Math StackExchange, Half iteration of exponential function.
Gottfried Helms, Coefficients for fractional iterates exp(x)-1.
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
Wikipedia, Functional square root.
EXAMPLE
0.498563...
MATHEMATICA
n = 23; p[s_, 1] := c[s]; p[0, n_] := c[0]^n; p[s_, n_] := p[s, n] = Expand[Sum[c[k] p[s - k, n - 1], {k, 0, s}]]; d[n_, 0] := Sum[c[k] c[0]^k, {k, 0, n}]; d[n_, i_] := Sum[c[k] p[i, k], {k, 1, n}]; a = c[0] /. FindRoot[Table[d[n, k] k! == 1, {k, 0, n}], Table[{c[k], 2^-k}, {k, 0, n}], WorkingPrecision -> 30]; First[RealDigits[a, 10, 6]]
CROSSREFS
KEYWORD
AUTHOR
Vladimir Reshetnikov, Nov 03 2011
STATUS
approved