A175739 Triangle T(n,m) with the coefficient [x^m] of the polynomial x^(2*n) - x^(2*n - 1) - x^n - x + 1 in row n, column m, 1 <= m <= 2*n. T(0,0) = 1.

1, 1, -3, 1, 1, -1, -1, -1, 1, 1, -1, 0, -1, 0, -1, 1, 1, -1, 0, 0, -1, 0, 0, -1, 1, 1, -1, 0, 0, 0, -1, 0, 0, 0, -1, 1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Roger L. Bagula, Dec 04 2010

Keywords

• sign
• easy
• tabf

Comments

The polynomials up to n = 10 are Salem polynomials (the third lowest Salem in the table).

Examples

			The polynomial coefficients are
  1;
  1, -3,  1;
  1, -1, -1, -1,  1;
  1, -1,  0, -1,  0, -1,  1;
  1, -1,  0,  0, -1,  0,  0, -1,  1;
  1, -1,  0,  0,  0, -1,  0,  0,  0, -1,  1;
  1, -1,  0,  0,  0,  0, -1,  0,  0,  0,  0, -1, 1;
  1, -1,  0,  0,  0,  0,  0, -1,  0,  0,  0,  0, 0, -1, 1;
  1, -1,  0,  0,  0,  0,  0,  0, -1,  0,  0,  0, 0,  0, 0, -1, 1;
  1, -1,  0,  0,  0,  0,  0,  0,  0, -1,  0,  0, 0,  0, 0,  0, 0, -1, 1;
  1, -1,  0,  0,  0,  0,  0,  0,  0,  0, -1,  0, 0,  0, 0,  0, 0,  0, 0, -1, 1;
  ...
The corresponding Mahler measures are
  -----------------------------------------------------
  n | M(p_n)                ||  n | M(p_n)
  -----------------------------------------------------
  1 | 1.7220838057390422450 ||  6 | 1.2612309611
  2 | 1.5061356795538388    ||  7 | 1.2363179318
  3 | 1.40126836793         ||  8 | 1.21639166113826509
  4 | 1.337313210201        ||  9 | 1.200026523
  5 | 1.293485953125        || 10 | 1.286735
  ...
		

Links

• Michael Mossinghoff, Small Salem Numbers
• William J. Floyd, Growth of planar Coxeter groups, P.V. numbers, and Salem numbers, Math. Ann. Vol. 293 (1992), 475-483.

Crossrefs

Cf. A143439.

Programs

Mathematica

p[x_, n_] = If[n == 0, 1, x^(2*n) - x^(2*n - 1) - x^n - x + 1];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

Maxima

T(n, k) := if n = 0 and k = 0 then 1 else ratcoef(x^(2*n) - x^(2*n - 1) - x^n - x + 1, x, k)$
create_list(T(n, k), n, 0, 10, k, 0, 2*n); /* Franck Maminirina Ramaharo, Nov 02 2018 */

Formula

Sum_{m=0..2*n} T(n,m)= -1.

From Franck Maminirina Ramaharo, Nov 02 2018: (Start)

G.f.: (1 - 4*x*y + x*(2 + x + 2*x^2)*y^2 - x^2*(1 + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - x^2*y)).

E.g.f.: (-(1 - x)*exp(x^2*y) - x*exp(x*y) + x*(1 - x)*exp(y) + 1 + x^2)/x. (End)