Formula for the Taylor Series for Tetration

[Catullus]

Does anyone know of a patterns for the terms of the Taylor series for tetration? (With the Kneser method.) Like does anyone know of any formula for the Taylor series for e4^^x? How could you find the 65536th term of the Taylor series, without having to differentiate 65536 times?

Hey Catullus....

I believe Fatou.gp has a manner of calculating taylor series! And you don't have to differentiate. Sheldon has a built in protocol. I've never used it, so I can't speak for its efficacy. I'm not sure if you'd be able to go that far out efficiently though...

This is also a reason I don't like sheldon's program for polynomial data. Every program I write will accept \(\text{sexp}(3+z)\) and spit out a polynomial to desired precision in \(z\), effectively solving the Taylor polynomial at \(3\). Unfortunately, Sheldon uses matrices and root finders built into his functions, so the values as input are required to be static (you can't plug in a polynomial into his functions, like I'd be able to).

The trouble is, of course, that Kneser is a slow process on its own. And there is no "convenient" formula for the polynomial. I had tried to write a program using the beta method which would estimate Kneser (and hence solve polynomial data and take polynomial arguments), but as I could never 100% prove the process works, I abandoned it. Plus, the code would probably run unreasonably slow compared to fatou.gp (god that program is fast).

Not sure what else to tell you. There's no convenient recursive formula for taylor terms, especially for Kneser.

What about for fraction iterations of exponentials?