How to prove the sequence of $p_n(x)$ are all integer polynomials on x?

for $x \in \mathbb R ,|x|< \frac12$ define:

$A_n(x)$ solution of the relation:

$A_{n+1}(x) = \frac{x^2}{1-A_n(x)}$

$A_0(x) = 0$

(I found that $A_n(x) = 2x^2 \cdot \frac{(1+\sqrt{1-4x^2})^n - (1 - \sqrt{1-4x^2})^n}{(1+\sqrt{1-4x^2})^{n+1} - (1 - \sqrt{1-4x^2})^{n+1}}$)

prove that the sequence:

$p_0(x)=1$

$p_{n+1}(x) = p_n(x) \cdot (1-x^2 - A_n(x))$

consists of integer polynomials in $x$

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