%I #41 Nov 27 2021 05:17:28
%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(n, x) = (p(n-1, x) + x)/(1 - x*p(n-1, x)), with p(0, x) = 0, p(1, x) = 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[n_, x_]:= p[n, x]= If[n<2, n*x, (p[n-1, x] + x)/(1 - x*p[n-1, x])];
%t Table[CoefficientList[Numerator[FullSimplify[p[n, x]]], x], {n,0,12}]//Flatten
%o (Sage)
%o 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 def A135685(n,k): return ( p(n,x) ).series(x,n+1).list()[k]
%o flatten([[A135685(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 26 2021
%Y Cf. A095704, A162590.
%K tabf,sign
%O 0,5
%A _Roger L. Bagula_, Feb 17 2008
%E Prepended first term and offset corrected by _James Burling_, Sep 14 2014