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Search: id:A152065
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| A152065 |
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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 - Floor[(n - 1)/2] - 1}] + 1/x, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - Floor[(n - 1)/2] - 1}] + 1]. |
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1
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| 1, 0, 1, 0, -1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, 0, -1, -1, -1, 1, 1, 0, -1, -1, -1, -1, 1, 1, 0, 0, -1, -1, -1, -1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 1, 1, 0, 0, 0, -1, -1, -1, -1, -1, 1, 1, 0, 0, 0, -1, -1, -1, -1, -1, -1, 1
(list; table; graph; listen)
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OFFSET
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0,1
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COMMENT
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These polynomials gives odd Salem polynomials starting with n=7. The row sums are: {1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4,...} Example: 1 - x^9 - x^10 - x^11 - x^12 - x^13 - x^14 - x^15 - x^16 - x^17 -x^18 + x^19; with absolute value roots: {1., 0.957624, 0.957624, 0.997081, 0.997081, 0.962514, 0.962514, 0.98743, 0.98743, 0.972887, 0.972887, 0.96672, 0.96672, 0.989308, 0.989308, 0.915352, 0.915352, 0.837413, 1.99902}.
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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 - Floor[(n - 1)/2] - 1}] + 1/x, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - Floor[(n - 1)/2] - 1}] + 1]; t(n,m/)=coefficients(p(x,n)).
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EXAMPLE
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{1}, {0, 1}, {0, -1, 1}, {1, -1, -1, 1}, {1, -1, -1, -1, 1}, {1,0, -1, -1, -1, 1}, {1, 0, -1, -1, -1, -1, 1}, {1, 0, 0, -1, -1, -1, -1, 1}, {1, 0, 0, -1, -1, -1, -1, -1, 1}, {1, 0, 0, 0, -1, -1, -1, -1, -1, 1}, {1, 0, 0, 0, -1, -1, -1, -1, -1, -1, 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 -Floor[(n - 1)/2] - 1}] + 1/x, x^n - x^Floor[(n - 1)/2]*Sum[x^m, {m, 0, n - Floor[(n - 1)/2] - 1}] + 1]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Sequence in context: A156728 A071039 A074332 this_sequence A113428 A133101 A145377
Adjacent sequences: A152062 A152063 A152064 this_sequence A152066 A152067 A152068
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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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