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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
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
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
Sequence in context: A218031 A135523 A194663 * A349447 A164658 A079067
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