Tetration series for integer exponent. Can you find the pattern?

[marraco]

I'm not that good on math, but....
I can help as a programmer.
Maybe precision problems are due to usage of "fixed" floating points (I mean fixed bit numbers).

I'm not even a mathematician. Just an engineer.

Pari/GP uses arbitrary precision, and I used like 300 or 400 decimals.
I also inverted the function for the calculation of the derivatives and verified that there is no loss of information.

...but beyond the precision used, the method of finite differences has inherent error. That may be the cause, or knesser may need more precision (with the one I choose, it took more than a day to run, but it may not be enough), or there is a bug on my posted code (ask for clarification if you want to check it), or all of those...