Something interesting about Taylor series

[bo198214]

http://mathworld.wolfram.com/PowerTower.html

formula ( 6 ) , ( 7 ) and ( 8 ).

a classic.

Hm, that is a development in \( \ln(x) \), Not really a Taylor devlopment of x^^n at some point \( x_0 \).
I wonder whether we have formulas for the powerseries development of \( x\^\^n \) at \( x_0=1 \) (and not at 0 because the powertower is not analytic there).

And indeed Andrew pointed it out in his tetration-reference formula (4.17-4.19) (or in the Andrew's older Tetration FAQ 20080112: (4.23-25)):

\( \begin{equation}
{}^{n}{x} = \sum^\infty_{k=0} t_{n,k} (x-1)^k
\end{equation} \)
where:
\( \begin{equation}
t_{n,k} = \begin{cases}
1 & \text{if } n \ge 0 \text{ and } k = 0, \\
0 & \text{if } n = 0 \text{ and } k > 0, \\
1 & \text{if } n = 1 \text{ and } k = 1, \\
0 & \text{if } n = 1 \text{ and } k > 1,
\end{cases}
\end{equation}
\)
otherwise:
\( \begin{equation}
t_{n,k} = \frac{1}{k} \sum^k_{j=1} \frac{1}{j} \sum^k_{i=j} i {(-1)^{j-1}} t_{n,k-i} t_{n-1,i-j}
\end{equation}
\)

I think it has convergence radius 1 because the substituted logarithm has convergence radius 1 and also because of the singularity at 0.