  We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and
$b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases
they can be written in closed form. Failing that, the sums still share many
common features: combinatorial interpretations, Pascal-like recurrences,
inverse relations with their signed versions, and interpretations as
coefficients of change between polynomial bases.
