A Olympiad problem (original creation)

[sirio11]

I have created many Olympiad problems, most of them for regional, state and National (Mexico) contests. This is not the one I'm proud the most, just one I find interesting and with a short solution.


(n,r) means nCr

Find all positive integers n,z,k that satisfy:

(n,0)^2+(n,1)^2+(n,2)^2+.....+(n,n)^2 = z^k