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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.)