The classical Stern sequence was extended by Klavazar, Milutinovic and Petr to the Stern polynomials Bn(x) defined by B0(x) = 0, B1(x) = 1, B2n(x) = xBn(x), and B2n+1(x) = Bn(x) + Bn+1(x). In this paper we prove several divisibility results for these polynomials. We also find several infinite classes of positive integers n such that the Stern polynomials with index n2 are squares of polynomials which we give explicitly. We conjecture that, apart from two sporadic square Stern polynomials, we have characterized them all.
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