# Evaluate Triple Sum

**Discrete Mathematics**Level 5

\[\large \displaystyle \sum _{ n=2 }^{ 2016 }{ \displaystyle \sum _{ k=0 }^{ n }{ \displaystyle \sum _{ j=0 }^{ \phi ( n ) +1 }{ \left(\dbinom nk \dfrac { { ( -1 ) }^{ n+k }{ ( n+k ) }^{ j } }{ ( \phi ( n ) +1 ) ! } \right) } } } = \, ? \]

For your final step of your calculation, you may refer to this list of prime numbers.

**Notations**:

- \(\phi(\cdot) \) denotes the Euler's totient function.
- \( \dbinom nk \) denotes the binomial coefficient, \( \dbinom nk = \dfrac{n!}{k!(n-k)!} \).
- \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).