  We use the holonomic ansatz to estimate the asymptotic behavior, in $T$, of
the average maximal number of balls in a bin that is obtained when one throws
uniformly at random (without replacement) $r$ balls into $n$ bins, $T$ times.
Our approach works, in principle, for any fixed $n$ and $r$. We were able to do
the cases $(n,r)$ = $(2,1),(3,1),(4,1), (4,2)$, but things get too complicated
for larger values of $n$ and $r$. We are pledging a \$150 donation to the OEIS
for an explicit expression, (in terms of $n$, $r$, and $\pi$) for the constant
$C_{n,r}$ such that that average equals $\frac{n}{r}\,T+C_{n,r}
\sqrt{T}+O(1/\sqrt{T})$. In this version we announce that the problem has been
solved (to the extent possible) by Marcus Michelen.
