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 A135685 Triangular sequence of the coefficients of the Numerator of the rational recursive sequence for tan(n*y) with x=tan(y). 1

%I

%S 0,0,1,0,-2,0,-3,0,1,0,4,0,-4,0,5,0,-10,0,1,0,-6,0,20,0,-6,0,-7,0,35,

%T 0,-21,0,1,0,8,0,-56,0,56,0,-8,0,9,0,-84,0,126,0,-36,0,1,0,-10,0,120,

%U 0,-252,0,120,0,-10,0,-11,0,165,0,-462,0,330,0,-55,0,1

%N Triangular sequence of the coefficients of the Numerator of the rational recursive sequence for tan(n*y) with x=tan(y).

%C Signed version of A034867 with interlaced zeros. - _Joerg Arndt_, Sep 14 2014

%C The negatives of these terms gives the coefficients for the numerators for when n is negative (i.e. tan(-n*y) = -tan(n*y)). - _James Burling_, Sep 14 2014

%H Robert Israel, <a href="/A135685/b135685.txt">Table of n, a(n) for n = 0..10082</a>

%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.

%F p(x,0)=0; p(x,1)=x; p(x, n) = (p(x, n - 1) + x)/(1 - p(x, n - 1)*x);

%F sum(j, T(n,j)*x^j) = g(n,x) where g(0,x) = 0, g(1,x) = x, g(n,x) = -2*(-1)^n*g(n-1,x) + (x^2+1)*g(n-2,x). - _Robert Israel_, Sep 14 2014

%e Triangle starts:

%e {0},

%e {0, 1},

%e {0, -2},

%e {0, -3, 0,1},

%e {0, 4, 0, -4},

%e {0, 5, 0, -10, 0, 1},

%e {0, -6, 0, 20, 0, -6},

%e {0, -7, 0, 35, 0, -21, 0,1},

%e {0, 8, 0, -56, 0, 56, 0, -8},

%e {0, 9, 0, -84, 0, 126, 0, -36, 0, 1},

%e {0, -10, 0, 120, 0, -252, 0, 120,0, -10},

%e {0, -11, 0, 165, 0, -462, 0, 330, 0, -55, 0, 1}

%p g[0]:= 0:

%p g[1]:= x;

%p for n from 2 to 20 do

%p g[n]:= expand(-2*(-1)^n*g[n-1]+(x^2+1)*g[n-2])

%p od:

%p 0, seq(seq(coeff(g[n],x,j),j=0..degree(g[n])),n=1..20); # _Robert Israel_, Sep 14 2014

%t p[x, 0] = 0; 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);

%t c = Table[CoefficientList[Numerator[FullSimplify[p[x, n]]], x], {n, 0, 11}];

%t Flatten[c]

%Y Cf. A095704, A162590.

%K tabf,sign

%O 0,5

%A _Roger L. Bagula_, Feb 17 2008

%E Prepended first term and offset corrected, _James Burling_, Sep 14 2014

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Last modified July 30 23:09 EDT 2021. Contains 346365 sequences. (Running on oeis4.)