Hi Marraco -
I'm currently unable to step in, and I do not yet understood your formulae. But there is some resemblance to some investigation I did years ago, and especially the topic "labelled rooted trees" reminded me of my treatize on the tetrated pascalmatrix, and of the links to the resp. entries in OEIS. Perhaps there is something in it for you; see this Pascalmatrix tetrated Perhaps the used procedure contains a method to compute the coefficients of T(x,k)... but might come out that the explicite computation of the matrix routines are eventually recursive; I don't know at the moment.
I'm currently unable to step in, and I do not yet understood your formulae. But there is some resemblance to some investigation I did years ago, and especially the topic "labelled rooted trees" reminded me of my treatize on the tetrated pascalmatrix, and of the links to the resp. entries in OEIS. Perhaps there is something in it for you; see this Pascalmatrix tetrated Perhaps the used procedure contains a method to compute the coefficients of T(x,k)... but might come out that the explicite computation of the matrix routines are eventually recursive; I don't know at the moment.
Gottfried Helms, Kassel
