Algebra Level 5

Let $$g(x)$$ denote a quintic $$\big(5^\text{th}$$-degree$$\big)$$ polynomial such that $$g(n) = 2^n$$ for $$n = 1,2,3,4,5$$ and $$g(x)$$ has a constant term of $$5!$$.

Find $$g(6)$$.

Bonus: Generalize this.

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