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The Stern polynomials defined by $s(0;x)=0$, $s(1;x)=1$, and for $n\geq 1$ by$s(2n;x)=s(n;x^2)$ and $s(2n+1;x)=x\,s(n;x^2)+s(n+1;x^2)$ have only 0 and 1 ascoefficients. We construct an infinite lower-triangular matrix related to thecoefficients of the $s(n;x)$ and show that its inverse has only 0, 1, and $-1$as entries, which we find explicitly. In particular, the sign distribution ofthe entries is determined by the Prouhet-Thue-Morse sequence. We also obtainother properties of this matrix and a related Pascal-type matrix that involvethe Catalan, Stirling, Fibonacci, Fine, and Padovan numbers. Further resultsinvolve compositions of integers, the Sierpi\'nski matrix, and identitiesconnecting the Stern and Prouhet-Thue-Morse sequences.

A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence