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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
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 (list; graph; refs; listen; history; text; internal format)
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(x,0)=0; p(x,1)=x; p(x, n) = (p(x, n - 1) + x)/(1 - p(x, n - 1)*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[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);

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

Flatten[c]

CROSSREFS

Cf. A095704, A162590.

Sequence in context: A218031 A135523 A194663 * A164658 A079067 A160271

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, James Burling, Sep 14 2014

STATUS

approved

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Last modified June 12 10:58 EDT 2021. Contains 344947 sequences. (Running on oeis4.)