I just found a nice "power series" for 
. It's based on regular iteration, and I took the regular iteration power series for )
and evaluated it at (z=1), then substituted (y=x) to obtain a power series for . At this point, I noticed there were lots of logarithms, or in other words, it seemed to be of the form ^k)
which is nice, but not very interesting. It was when I noticed that the power series expansion of 
about 0 also produces this kind of series that I thought the two could be combined to make a new kind of series. So I tried doing some linear algebra change-of-basis stuff but I forgot how, so I used "undetermined coefficients" instead.
I want to discuss this more, but it will have to wait until this weekend. Here is the power series.
 \uparrow\uparrow (x + 1) <br />
& = 1 \\<br />
& + x \\<br />
& + x^2 (1 - x^x) \\<br />
& + \frac{x^3}{2}(3 - 3 x^x) \\<br />
& + \frac{x^4}{3!}(14 - 17 x^x) \\<br />
& + \frac{x^5}{4!}(96 - 119 x^x - 12 x^{2x}) \\<br />
& + \frac{x^6}{5!2}(1698 - 2013 x^x - 420 x^{2x}) \\<br />
& + \frac{x^7}{6!4}(37448 - 41199 x^x - 15480 x^{2x}) \\<br />
& + \cdots<br />
\end{tabular}<br />
)
[update]Fixed some coefficient signs[/update]
Andrew Robbins
I want to discuss this more, but it will have to wait until this weekend. Here is the power series.
[update]Fixed some coefficient signs[/update]
Andrew Robbins
