Number of Distinct Values of \(f(n)\)

Algebra Level 4

\[\large f(n) = \sum_{k =0}^{n}\frac{(-1)^{n+k}(k+1)^n}{k!(n-k)!}\]

Let \(n\) be a positive integer, find the number of distinct values of \(f(n) \).

Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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