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A036279 Denominators in the Taylor series for tan(x).

%I A036279 #63 Feb 16 2025 08:32:37
%S A036279 1,3,15,315,2835,155925,6081075,638512875,10854718875,1856156927625,
%T A036279 194896477400625,2900518163668125,3698160658676859375,
%U A036279 1298054391195577640625,263505041412702261046875,122529844256906551386796875,4043484860477916195764296875
%N A036279 Denominators in the Taylor series for tan(x).
%C A036279 The n-th coefficient of Taylor series for tan(x) appears to be identical to the quotient A160469(n)/A156769(n). - _Johannes W. Meijer_, May 24 2009
%D A036279 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.67).
%D A036279 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%D A036279 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A036279 A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 74.
%D A036279 H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 329.
%D A036279 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, equation 34:6:1 at page 323.
%H A036279 Seiichi Manyama, <a href="/A036279/b036279.txt">Table of n, a(n) for n = 1..253</a> (first 100 terms from T. D. Noe)
%H A036279 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A036279 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.67).
%H A036279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperbolicTangent.html">Hyperbolic Tangent</a>.
%H A036279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Tangent.html">Tangent</a>.
%H A036279 Herbert S. Wilf, <a href="https://www2.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, Academic Press, NY, 1990. See p. 50.
%F A036279 a(n) = denominator((-1)^(n-1)*2^(2*n)*(2^(2*n)-1)*Bernoulli(2*n)/(2*n)!). - _Johannes W. Meijer_, May 24 2009
%F A036279 Let R(x) = (cos(x*Pi/2) + sin(x*Pi/2))*(4^x-2^x)*Zeta(1-x)/(x-1)!. Then a(n) = denominator(R(2*n)) and A002430(n) = numerator(R(2*n)). - _Peter Luschny_, Aug 25 2015
%e A036279 tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + ... = x + (1/3)*x^3 + (2/15)*x^5 + (17/315)*x^7 + (62/2835)*x^9 + ... =
%e A036279   Sum_{n >= 1} (2^(2n) - 1) * (2x)^(2n-1) * |Bernoulli_2n| / (n*(2n-1)!).
%e A036279 The coefficients in the expansion of tan x are 0, 1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835, 0, 1382/155925, 0, 21844/6081075, 0, 929569/638512875, 0, ... = A002430/A036279
%e A036279 tanh(x) = x - (1/3)*x^3 + (2/15)*x^5 - (17/315)*x^7 + (62/2835)*x^9 - (1382/155925)*x^11 + ...
%e A036279 The coefficients in the expansion of tanh x are 0, 1, 0, -1/3, 0, 2/15, 0, -17/315, 0, 62/2835, 0, -1382/155925, 0, 21844/6081075, 0, -929569/638512875, 0, 6404582/10854718875, 0, -443861162/1856156927625, ... = A002430/A036279
%p A036279 R := n -> (-1)^floor(n/2)*(4^n-2^n)*Zeta(1-n)/(n-1)!:
%p A036279 seq(denom(R(2*n)), n=1..18); # _Peter Luschny_, Aug 25 2015
%t A036279 f[n_] := (-1)^Floor[n/2] (4^n - 2^n) Zeta[1 - n]/(n - 1)!; Table[Denominator@ f[2 n], {n, 17}] (* _Michael De Vlieger_, Aug 25 2015 *)
%Y A036279 Cf. A002430, A000182, A099612, A156769, A160469.
%K A036279 nonn,easy,frac
%O A036279 1,2
%A A036279 _N. J. A. Sloane_
%E A036279 Incorrect comment by Stephen Crowley deleted by _Johannes W. Meijer_, Jan 19 2009