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A036279 Denominators in Taylor series for tan(x). 13
1, 3, 15, 315, 2835, 155925, 6081075, 638512875, 10854718875, 1856156927625, 194896477400625, 2900518163668125, 3698160658676859375, 1298054391195577640625, 263505041412702261046875, 122529844256906551386796875, 4043484860477916195764296875 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The Taylor series for tan(x) appears to be identical to the quotient of the "look-alikes" of the numerator and denominator, i.e., A160469(n)/A156769(n). - Johannes W. Meijer, May 24 2009

REFERENCES

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).

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

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

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.

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

LINKS

T. D. Noe and Seiichi Manyama, Table of n, a(n) for n = 1..253 (first 100 terms from T. D. Noe)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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).

Eric Weisstein's World of Mathematics, Hyperbolic Tangent

Eric Weisstein's World of Mathematics, Tangent

FORMULA

a(n) = denom((-1)^(n-1)*2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n)!). - Johannes W. Meijer, May 24 2009

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

EXAMPLE

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 + ... =

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

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

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

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

MAPLE

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

seq(denom(R(2*n)), n=1..18); # Peter Luschny, Aug 25 2015

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A138896 A090627 A070234 * A156769 A029758 A103031

Adjacent sequences:  A036276 A036277 A036278 * A036280 A036281 A036282

KEYWORD

nonn,easy,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified March 30 20:18 EDT 2017. Contains 284302 sequences.