I was looking at tetration and considering its little taylor series, given by:

(1)

where:

and:

specifically, what I was trying to do is find a relationship between the coefficients of tetrations' little taylor series and normal taylor series. We'll say:

so I'm looking for as an expression of or vice versa.

So I started off by looking at (1):

and now let this equal our formula for sexp(x+1) using or the normal taylor series.

and subtract the right hand side from the left hand side

and now, since x is essentially arbitrary, let x = l to give:

but this contradicts the original Taylor series expansion

which states:

Any help would be greatly appreciated, thanks.

The only solution I have is

(1)

where:

and:

specifically, what I was trying to do is find a relationship between the coefficients of tetrations' little taylor series and normal taylor series. We'll say:

so I'm looking for as an expression of or vice versa.

So I started off by looking at (1):

and now let this equal our formula for sexp(x+1) using or the normal taylor series.

and subtract the right hand side from the left hand side

and now, since x is essentially arbitrary, let x = l to give:

but this contradicts the original Taylor series expansion

which states:

Any help would be greatly appreciated, thanks.

The only solution I have is