I have a feeling that the disagreement might come from the fact that computer software systems, being fed this formula for $k=n$, encounter the summand $\binom{-1}{k-1}$ which they evaluate as $(-1)^{k-1}$ (which is not what you and me assume), hence the discrepancy.

Well, the convention I would definitely like to follow is $\binom{p}{q}\ne 0$ only for $0\le q\le p$, and in this case is given by the usual formula.

@Pietro: I believe my formula gives 1520 as well: $1716-7\cdot 1-21\cdot 4-35\cdot3=1520$.

Wadim, over $\mathbb{Z}$ the number is $8(p+1)$, so I assume Roland is correct (since these primes cannot be represented by less than 4 squares, for positive solutions we have to divide by 16)...

True - I keep forgetting about the built-in nonlinearity :-)

Victor, does not one of the sentences in Popov's article read "A linear algebraic group over a field of characteristic 0 is reductive if and only if its Lie algebra is a reductive Lie algebra"? It is more than an bit misleading, isn't it?

@Victor: I believe that at some point one of the existing versions of terminology might have been "reductive" for Lie groups with reductive Lie algebras and "linearly reductive" for what now is more commonly called reductive. Some traces of that mess remained till today, see, e.g. eom.springer.de/R/r080440.htm (which messes up as many things as possible) and eom.springer.de/l/l058500.htm. But it's probably true that this terminology is mostly dormant, and is more or less non-existent in reliable textbooks.

@Michele: that's fine, a good lesson for me to read things carefully. A reductive algebraic group is reductive as a Lie group, sure, but I fail to see how it is relevant: for a reductive Lie group, the statement you are interested in is false, while for a reductive algebraic group it is true, so you'd better be careful too :)

@Victor: thanks! yes, I guess I read the title of this question better than the body :( I edited the answer accordingly.

@Ben: your comment really confuses me... since for a basis of the free Lie algebra we can take Lyndon words (a very combinatorial object), the above formula can be thought of as a sum of one term which has a clear combinatorial meaning!