A152066 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].

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

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.

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

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: A188641 A086299 A131364 * A334946 A225595 A228813

Adjacent sequences: A152063 A152064 A152065 * A152067 A152068 A152069

Keyword

tabf,sign,uned

Author

Roger L. Bagula, Nov 23 2008

Status

approved