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

[Gottfried]

That's great. You found that the tetration of the pascal matrix produces the bj sequences.
If we can iterate logaritms maybe we can figure what a tree of negative height is.

I wonder if we can generate the tetrated pascal with a series, and if that series matches the series for some real base. That base should be important.

If just we could calculate \( \\[15pt]

{^{1.5}P} \), we would get the bj for trees of 1/2 height.

Hi marraco -

the point of the logarithm L of the Pascalmatrix P and its non-invertibility is likely not the end of the story here. Although it is only one subdiagonal below the diagonal and the diagonal is empty/is zero this is the same situation as we face it with the fractional derivative, expressed in the matrix form. There the derivative-operator is exactly that subdiagonal-matrix. And the second-derivative-operator is its second power and so on. But anyway we can work with somehow "the inverse", namely the integration. That operator has the upper subdiagonal populated with values. One of the reasons for the possibility to step forwards is likely the fact of the infinite size of the matrices which allow sometimes operations which are impossible/contradictory with finite sized-matrices.

And finally, there is also the concept of fractional derivatives (in fact a complete handful of concepts) which indeed define a squareroot of that derivative-operator or subdiagonal-matrix. However it must be made working seriously and I feel a bit tired to dive into this -for me new- concept. I tried to become more familiar with it and, for instance got stuck with the question how to define the half-derivative of the zeta at zero (based on the Dirichlet-series representation) and this question is still open in mathoverflow... So because I also had not much time besides my teachings this still lingers around - and if you like that whole question more and like try experimentate with that matrix-operators in Pari/GP you can also get my set of routines (which can help to make the matrix-things better visible - for instance all the nice matrix-pictures in my article are made with the PariTTY-GUI for Pari/GP). (You better email me privately for this because I surely must explain in detail this and that and much and more...)

Gottfried