  In Gert Almkvist's beautiful article, entitled "Invariants, mostly old ones",
(that appeared in the Pacific Journal of Mathematics, vol. 86 (1980), pp. 1-13)
he talked about a sequence of generating functions that came up in his work,
that turned out to be the same as generating functions for the number of
covariants of binary quadratic forms studied by Faa de Bruno, Cayley,
Sylvester, and other 19th century savantes. Using a very simple-minded Maple
program (that uses the partial-fraction decomposition of a rational function),
we recompute them, and go all the way to degree 18. It turns out that the same
method can be used to answer many other enumeration questions, including the
one in the title.
