A152065 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].

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

Offset

0,1

Comments

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}.

Links

Table of n, a(n) for n=0..65.

Formula

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)).

Example

{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}

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 -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]

Crossrefs

Sequence in context: A331990 A301849 A074332 * A267870 A267878 A231367

Adjacent sequences: A152062 A152063 A152064 * A152066 A152067 A152068

Keyword

tabl,sign

Author

Roger L. Bagula, Nov 23 2008

Status

approved