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Irregular triangle of polynomial coefficients: p(x,n)=If[n == 0, x^n - x^Floor[(n - 1)/ 2]*Sum[x^m, {m, 0, n - 2*Floor[(n - 1)/ 2]}] + 1/x, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - 2*Floor[(n - 1)/2]}] + 1].
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%I #9 Feb 13 2023 02:59:13

%S 1,-1,-1,1,1,-1,-1,-1,1,1,0,-1,-1,0,1,1,0,-1,-1,-1,0,1,1,0,0,-1,-1,0,

%T 0,1,1,0,0,-1,-1,-1,0,0,1,1,0,0,0,-1,-1,0,0,0,1,1,0,0,0,-1,-1,-1,0,0,

%U 0,1

%N Irregular triangle of polynomial coefficients: p(x,n)=If[n == 0, x^n - x^Floor[(n - 1)/ 2]*Sum[x^m, {m, 0, n - 2*Floor[(n - 1)/ 2]}] + 1/x, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - 2*Floor[(n - 1)/2]}] + 1].

%C These polynomials give Salem polynomials starting with n=3 and ending with 12.

%C The row sums are: {-1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1,...}

%C Example: 1 - x^5 - x^6 - x^7 + x^12; with absolute value roots: {1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 0.850137, 1.17628}.

%e {1, -1, -1, 1},

%e {1, -1, -1, -1, 1},

%e {1, 0, -1, -1, 0, 1},

%e {1, 0, -1, -1, -1, 0, 1},

%e {1, 0, 0, -1, -1, 0, 0, 1},

%e {1, 0, 0, -1, -1, -1, 0, 0, 1},

%e {1, 0, 0, 0, -1, -1, 0, 0, 0, 1},

%e {1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 1},

%t p[x_, n_] = If[n == 0, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - 2*Floor[(n - 1)/2]}] + 1/x, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - 2*Floor[(n - 1)/2]}] + 1];

%t Table[ExpandAll[p[x, n]], {n, 3, 10}];

%t a = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 3, 10}];

%t Flatten[a]

%K tabf,sign,uned

%O 3,1

%A _Roger L. Bagula_, Nov 23 2008