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A152066 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]. 0
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; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

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}.

LINKS

Table of n, a(n) for n=3..62.

FORMULA

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)).

EXAMPLE

{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}

MATHEMATICA

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]

CROSSREFS

Sequence in context: A105586 A136522 A086299 * A122255 A122261 A014922

Adjacent sequences:  A152063 A152064 A152065 * A152067 A152068 A152069

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Nov 23 2008

STATUS

approved

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Last modified July 27 04:25 EDT 2017. Contains 289841 sequences.