OBM 2014 Problem 4

The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as $$P_0(x)=x, P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1), n\ge 1.$$ Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$ .