IR-TST 2017 Problem 17

Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$ . A sequence of polynomials is defined as $$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$ Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that $$P_{2n}(x)=P_n(x^2+c).$$ Proposed by Navid Safaei