I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely
$$ \small f_p(x) = \sum_{k=0}^{\infty} \log(k!)^p \cdot x^k $$
and my first question is to find closed forms for $\small f_p(-1) $ for consecutive p.
They are all non-converging series, but which can nicely be summed by, for instance, Euler-summation, but I wish to find closed forms (or simple forms of other series with more known analytical properties). For p=0 this is $\small \eta(0) $ (the "Dirichlet's eta", or "alternating zeta", function), for p=1 this is $\small \eta(0)' $ and I expected, that $\small f_2(-1) $ would be something composed by the square of $\small f_1(-1)$ or the second derivative of the $\small \eta $ at zero, but didn't succeed so far.
The numerical values for the first few p seem to be (using Pari/GP sumalt-procedure)
$\small \qquad
\begin{array} {rl}
p & f_p(-1) \\
\hline \\
0 & 0.500000000000 \\
1 & 0.112895676322 \\
2 & -0.0380319653072 \\
3 & 0.0135052530749 \\
4 & 0.0183298626301 \\
5 & -0.107164190642 \\
6 & 0.331363715855 \\
7 & -0.482387386451 \\
8 & -2.91602127867 \\
9 & 32.5904726686 \\
10 & -154.360744999 \\
11 & -162.033212532\\
\end{array} $
