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

[Gottfried]

\( ^0(1+x) \,=\, \)\( {\color{Red} 1} \)\( + 0+ 0+0... \)

\( ^1(1+x) \,=\, \)\( {\color{Red} 1}+ x \)\( + 0+0+0... \)

\( ^2(1+x) \,=\, \)\( {\color{Red} 1}+ x+ x^2 \)\( +\frac {1}{2}x^{3}+\frac {1}{3}x^{4}+\frac {1}{12}x^{5}+\frac {3}{40}x^{6}+\frac {-1}{120}x^{7}+\frac {59}{2520}x^{8}+\frac {-71}{5040}x^{9}+\frac {131}{10080}x^{10}+\frac {-53}{5040}x^{11}+O(x^{12}) \)

\( ^3(1+x) \,=\, \)\( {\color{Red} 1}+x+x^2 +\frac {3}{2}x^{3} \)\( +\frac {4}{3}x^{4}+\frac {3}{2}x^{5}+\frac {53}{40}x^{6}+\frac {233}{180}x^{7}+\frac {5627}{5040}x^{8}+\frac {2501}{2520}x^{9}+\frac {8399}{10080}x^{10}+\frac {34871}{50400}x^{11}+O(x^{12}) \)

Perhaps this is interesting for you: http://go.helms-net.de/math/tetdocs/Pasc...trated.pdf The tetration of the Pascalmatrix give that coefficients by matrix-exponentiation, and from this might possibly result smoother formulae for the expression of the single coefficients.

Gottfried