OFFSET
0,3
COMMENTS
The polynomials up to n = 10 are Salem polynomials (the third lowest Salem in the table).
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.
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)
EXAMPLE
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
...
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
PROG
(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 */
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Roger L. Bagula, Dec 04 2010
STATUS
approved