Question: summation (not addition) over even/odd pos integers?

Maple evaluates

> sum(1/z^2, z = 1 .. infinity)

to (1/6)*Pi^2. What do I do when sumation is over all even positive integers? Is there any  closed-form symbolic formula for this case?

Above is the minimal example. What I need is to compute 

> sum(2*binomial(m-1, k-1)*binomial(n-1, k-1)/binomial(m+n, m), k = 1 .. infinity)

where k belongs to a set of only even positive integers, not any posint. The second expression evaluates to 2*m*n/((m+n-1)*(m+n)) and I wonder what happens when summation is over even posints only?

I don't have any problems with add(), e.g.,

> feven := proc (N, n, m) local r; add(`if`(type(r, even), 2*binomial(m-1, (1/2)*r-1)*binomial(n-1, (1/2)*r-1)/binomial(m+n, m), 0), r = 1 .. N) end proc

but I guess what I want is sum() and N=infty?