\[\large{ \begin{cases} \displaystyle f(n) = \sum_{k=1}^n \sum_{i=1}^{k+1} \dfrac{\binom{k}{i-1}^2 \binom{2k}{k} }{2^{2k} \binom{2k}{2i-2} (2i-1)} \\ \text{and} \\ \displaystyle g(n) = \sum_{k=1}^n \sum_{i=1}^{k+1} \dfrac{\binom{k}{i-1}^2 \binom{2k}{k} }{2^{2k} \binom{2k}{2i-2} i} \end{cases} } \]

Find the value of \(f(20) + g(15) \) upto three correct places of decimals.

**Bonus**: Generalize \(f(n)\) and \(g(n)\).

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