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A036279 Denominators in Taylor series for tan(x). 13

%I

%S 1,3,15,315,2835,155925,6081075,638512875,10854718875,1856156927625,

%T 194896477400625,2900518163668125,3698160658676859375,

%U 1298054391195577640625,263505041412702261046875,122529844256906551386796875,4043484860477916195764296875

%N Denominators in Taylor series for tan(x).

%C The Taylor series for tan(x) appears to be identical to the quotients A160469(n)/A156769(n). - _Johannes W. Meijer_, May 24 2009

%D 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 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.

%D 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 H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 329.

%H T. D. Noe and 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicTangent.html">Hyperbolic Tangent</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tangent.html">Tangent</a>

%F a(n) = denom((-1)^(n-1)*2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n)!). - _Johannes W. Meijer_, May 24 2009

%F 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 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 Sum_{n >= 1} (2^(2n) - 1) * (2x)^(2n-1) * |bernoulli_2n| / (n*(2n-1)!).

%e 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 tanh(x) = x - 1/3*x^3 + 2/15*x^5 - 17/315*x^7 + 62/2835*x^9 - 1382/155925*x^11 + ...

%e 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 R := n -> (-1)^floor(n/2)*(4^n-2^n)*Zeta(1-n)/(n-1)!:

%p seq(denom(R(2*n)), n=1..18); # _Peter Luschny_, Aug 25 2015

%t 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 Cf. A002430, A000182, A099612, A156769, A160469.

%K nonn,easy,frac

%O 1,2

%A _N. J. A. Sloane_

%E I deleted the comment by Stephen Crowley. His formula leads to incorrect values for higher values of this series. - _Johannes W. Meijer_, Jan 19 2009

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Last modified August 19 15:55 EDT 2018. Contains 313878 sequences. (Running on oeis4.)