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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).

%I A135685 #41 Nov 27 2021 05:17:28
%S A135685 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 A135685 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 A135685 0,-252,0,120,0,-10,0,-11,0,165,0,-462,0,330,0,-55,0,1
%N A135685 Triangular sequence of the coefficients of the numerator of the rational recursive sequence for tan(n*y) with x = tan(y).
%C A135685 Signed version of A034867 with interlaced zeros. - _Joerg Arndt_, Sep 14 2014
%C A135685 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 A135685 Robert Israel, <a href="/A135685/b135685.txt">Table of n, a(n) for n = 0..10082</a>
%H A135685 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 A135685 p(n, x) = (p(n-1, x) + x)/(1 - x*p(n-1, x)), with p(0, x) = 0, p(1, x) = x.
%F A135685 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 A135685 Triangle starts:
%e A135685   0;
%e A135685   0,   1;
%e A135685   0,  -2;
%e A135685   0,  -3,  0,   1;
%e A135685   0,   4,  0,  -4;
%e A135685   0,   5,  0, -10,  0,    1;
%e A135685   0,  -6,  0,  20,  0,   -6;
%e A135685   0,  -7,  0,  35,  0,  -21,  0,   1;
%e A135685   0,   8,  0, -56,  0,   56,  0,  -8;
%e A135685   0,   9,  0, -84,  0,  126,  0, -36,  0,   1;
%e A135685   0, -10,  0, 120,  0, -252,  0, 120,  0, -10;
%e A135685   0, -11,  0, 165,  0, -462,  0, 330,  0, -55,  0,  1;
%p A135685 g[0]:= 0:
%p A135685 g[1]:= x;
%p A135685 for n from 2 to 20 do
%p A135685 g[n]:= expand(-2*(-1)^n*g[n-1]+(x^2+1)*g[n-2])
%p A135685 od:
%p A135685 0, seq(seq(coeff(g[n],x,j),j=0..degree(g[n])),n=1..20); # _Robert Israel_, Sep 14 2014
%t A135685 p[n_, x_]:= p[n, x]= If[n<2, n*x, (p[n-1, x] + x)/(1 - x*p[n-1, x])];
%t A135685 Table[CoefficientList[Numerator[FullSimplify[p[n, x]]], x], {n,0,12}]//Flatten
%o A135685 (Sage)
%o A135685 def p(n, x): return n*x if (n<2) else 2*(-1)^(n+1)*p(n-1,x) + (1+x^2)*p(n-2,x)
%o A135685 def A135685(n,k): return ( p(n,x) ).series(x,n+1).list()[k]
%o A135685 flatten([[A135685(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 26 2021
%Y A135685 Cf. A095704, A162590.
%K A135685 tabf,sign
%O A135685 0,5
%A A135685 _Roger L. Bagula_, Feb 17 2008
%E A135685 Prepended first term and offset corrected by _James Burling_, Sep 14 2014