The r-power "binomial moments" are f(k,r) = sum_p binomial(2*k, k+p) * |p|^r and for any fixed r=0,1,2,... can be evaluated in closed form, see http://arxiv.org/abs/math/0606080 (and also MAPLE9 is capable of doing these for r=0,1,2,...,9 with some human help). I instead want to discuss some DOUBLE sums defining a different kind of r-power moment. Let f(k,s,r) = sum_p sum_q binomial(2*k, k+p)*binomial(2*k, k+q)*|p^2+s*q^2|^r where I have in mind that s=+1 or s=-1, and r=0,1,2,3... The sums are over all integers p,q, or if you want to be pedestrian then for -k <= p <= k, -k <= q <= k. If s=+1 or if r=even then the absolute value sign is obviated whereupon the double sums are expressible in terms of the above single-sums, no problemo. Life only is difficult if r=odd and s=-1. But in those cases Richard Brent & I surprised ourselves by discovering various closed forms: f(k, -1, 1) = 2 * k^2 * binomial(2*k,k)^2 f(k, -1, 3) = 8*binomial(2*k-2,k-1)^2 * k * (2*k-1) * (8*k^2-12*k+5) f(k, -1, 5) = 8*binomial(2*k-2,k-1)^2 * k * (256*k^5-1152*k^4+2112*k^34-1936*k^2+874*k-153) ...except I do not know how to prove these. (They are empirically correct.) Brent thinks he can prove the first, but I'll believe his proof when I see it :) So, what is the general formula and how to prove it?