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Function $r(n,k)$ and thus $g(n,k)$ can be computed from $f(n,k)$ as follows:

var('k n')
f(n,k) = factorial(n)^4 / factorial(k)^2 / factorial(n-k)^2 / factorial(2*n)
print( f.WZ_certificate(n, k) )

which prints 1/2*(2*k - 3*n - 3)*k^2/((k - n - 1)^2*(2*n + 1)) matching given $r(n,k)$.