A Confused Equation!

Algebra Level 4

\[\large{\dfrac{\binom{n}{n-1}^6 + \binom{n-2}{k}^6 + \binom{n+3}{n+1}^3 }{3 \binom{n-2}{k}^2 \binom{n+3}{2} } = n^2}\]

Determine the sum of all positive integers \(n,k\) for which the above equation satisfies.

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