The Power Series definition for tetration

[bo198214]

Can anyone express the nested series for the tetration powers of 2 and higher as just a single power series?

Ioannis Galidakis can. He gave this recurrence equation in his paper:
\( x{\uparrow}{\uparrow}n = \sum_{k=0}^{\infty} p_{nk} \ln(x)^k \) where \( p_{nk} = \frac{1}{k} \sum_{j=1}^{k} j p_{n(k-j)} p_{(n-1)(j-1)} \)
for more information, please see section 4.2.3 (page 26) in the Tetration Reference

Andrew wrote down there also the direct series development at (fixed point) x=1.