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

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

Offset

0,5

Comments

Signed version of A034867 with interlaced zeros. - Joerg Arndt, Sep 14 2014

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

Links

Robert Israel, Table of n, a(n) for n = 0..10082

Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.

Formula

p(n, x) = (p(n-1, x) + x)/(1 - x*p(n-1, x)), with p(0, x) = 0, p(1, x) = x.

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

Example

Triangle starts:
  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,  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,  0, -252,  0, 120,  0, -10;
  0, -11,  0, 165,  0, -462,  0, 330,  0, -55,  0,  1;

Maple

g[0]:= 0:
g[1]:= x;
for n from 2 to 20 do
g[n]:= expand(-2*(-1)^n*g[n-1]+(x^2+1)*g[n-2])
od:
0, seq(seq(coeff(g[n], x, j), j=0..degree(g[n])), n=1..20); # Robert Israel, Sep 14 2014

Mathematica

p[n_, x_]:= p[n, x]= If[n<2, n*x, (p[n-1, x] + x)/(1 - x*p[n-1, x])];
Table[CoefficientList[Numerator[FullSimplify[p[n, x]]], x], {n, 0, 12}]//Flatten

Prog

(Sage)
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)
def A135685(n, k): return ( p(n, x) ).series(x, n+1).list()[k]
flatten([[A135685(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 26 2021

Crossrefs

Cf. A095704, A162590.

Sequence in context: A218031 A135523 A194663 * A349447 A164658 A079067

Adjacent sequences: A135682 A135683 A135684 * A135686 A135687 A135688

Keyword

tabf,sign

Author

Roger L. Bagula, Feb 17 2008

Extensions

Prepended first term and offset corrected by James Burling, Sep 14 2014

Status

approved