In [reduce](http://sourceforge.net/projects/reduce-algebra/), I typed
~~~~
load zeilberg;
summand:=k*(k+1)*(2*k+1)*factorial(n)*factorial(n-1)/(factorial(n-k)*factorial(n+k+1));
gosper(summand,k,1,n);
~~~~
which resulted in
$$ \frac{(n+1)(n-1)!n}{(n+1)!}$$
which is a funny way of writing 1. I don't feel much wiser...