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Search: id:A152066
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| A152066 |
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A triangular sequence 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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1
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| 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, 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, 0, 1
(list; table; graph; listen)
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OFFSET
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3,1
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COMMENT
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These polynomials give Salem polynomials starting with n=3 and ending with 12. The row sums are: {-1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1,...} 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}.
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FORMULA
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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(n,m/)=coefficients(p(x,n)).
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EXAMPLE
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{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, 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, 0, 1}
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MATHEMATICA
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Clear[p, x, n, a, m]; 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]; Table[ExpandAll[p[x, n]], {n, 3, 10}]; a = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 3, 10}]; Flatten[a]
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CROSSREFS
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Sequence in context: A105586 A136522 A086299 this_sequence A122255 A122261 A014922
Adjacent sequences: A152063 A152064 A152065 this_sequence A152067 A152068 A152069
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KEYWORD
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tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 23 2008
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