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...

~~~~

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...