Let $P(x)$ be a polynomial of degree 2015 which satisfies $P(k) = 2^{k}$ for all $k=0,1,2,3, \ldots, 2015$. If the value of $P(2016)$ can be expressed as

$a^{b} - c,$

where $a$ and $c$ are coprime integers and $a$ is a prime, find the value of $a+b+c$.