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%I #8 Dec 07 2016 10:30:26
%S 1,1,-1,0,1,-1,0,3,1,0,-6,0,1,1,0,-10,0,5,-1,0,15,0,-15,0,1,-1,0,21,0,
%T -35,0,7,1,0,-28,0,70,0,-28,0,1,1,0,-36,0,126,0,-84,0,9,-1,0,45,0,
%U -210,0,210,0,-45,0,1,-1,0,55,0,-330,0,462,0,-165,0,11
%N Triangular sequence of the coefficients of the denominator of the rational recursive sequence for tan(n*x).
%C These are the denominators of the expansion of tan(n*x) as in A034839, but keeping the zeros with the terms in the denominator polynomials that vanish. Sign conventions differ slightly, maintaining either a positive coefficient [x^0], or a positive coefficient [x^n] or [x^(n-1)], resp.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kimberling/kimberling56.html">Polynomials associated with reciprocation</a>, JIS 12 (2009) 09.3.4, section 5.
%e {1},
%e {1},
%e {-1, 0, 1},
%e {-1, 0, 3},
%e {1, 0, -6,0, 1},
%e {1, 0, -10, 0, 5},
%e {-1, 0, 15, 0, -15, 0, 1},
%e {-1, 0, 21, 0, -35, 0, 7},
%e {1, 0, -28, 0, 70, 0, -28, 0, 1},
%e {1, 0, -36,0, 126, 0, -84, 0, 9},
%e {-1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1},
%e {-1, 0, 55, 0, -330, 0, 462, 0, -165, 0, 11}
%t Clear[p, x, a, b] p[x, 0] = 1; p[x, 1] = x; p[x, 2] = 2*x/(1 - x^2); p[x, 3] = (3*x - x^3)/(1 - 3*x^2); p[x_, n_] := p[x, n] = (p[x, n - 1] + x)/(1 - p[x, n - 1]*x); c = Table[CoefficientList[Denominator[FullSimplify[p[x, n]]], x], {n, 0, 11}]; Flatten[c]
%K sign,frac
%O 0,8
%A _Roger L. Bagula_, Feb 17 2008
%E Edited by the Associate Editors of the OEIS, Aug 18 2009
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