[seqfan] A019575 challenge

[Charles Greathouse]

A019575 is:
Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) =
number of ways that max in any box is k, for 1<=k<=n; sequence gives
triangle of numbers T(n,k).

It strikes me as a useful sequence that comes up in many applications.
 For example, I recently found someone asking (not in these terms) for
a weighted sum over T(16, k) for the purpose of estimating the number
of cache misses in a GPU.

The formula, as listed, is confusing.  Can anyone determine a better
(ideally, an efficient) formula or gf?  Also, a b-file might be nice,
though at the moment there's no good way to visualize large triangular
arrays.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University