We have $k$ different objects and $n$ different boxes. In how many ways can we put those objects into boxes so that each box gets 0, 1 or 2 objects? Each object has to be in a box.

This looks different than any of the standard combinatorial problems (like partitions etc). It's relatively easy to express this as a sum or to get some kind of recursive formula, but can we get a closed formula?

differentobjects, which I take to mean that they're distinguishable (note that he used exactly the same adjective for the boxes). The sequence you quote assumes the objects are indistinguishable. In that case the sequence in Barry Cipra's comment is certainly correct for the case $n=3$. – Steven Landsburg Mar 10 '13 at 16:10