\[\large{ S_N(n) = \displaystyle \sum_{k \geq n} \binom{N}{2k} \binom{k}{n} }\]

Let \(S_N(n)\) be the combinatorial summation as described above for integers \(n \geq 0\) and \(N \geq 1\). Generalize \(S_N(n)\) in terms of \(N\) and \(n\). Then using a calculator, evaluate the value of \(S_{40}(15)\).

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