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A175739
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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.
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5
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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
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OFFSET
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0,3
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COMMENTS
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The polynomials up to n = 10 are Salem polynomials (the third lowest Salem in the table).
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LINKS
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FORMULA
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Sum_{m=0..2*n} T(n,m)= -1.
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)
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EXAMPLE
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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
...
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MATHEMATICA
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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
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PROG
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(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)$
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CROSSREFS
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KEYWORD
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sign,easy,tabf
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AUTHOR
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STATUS
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approved
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