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

1, 3, 15, 315, 2835, 155925, 6081075, 638512875, 10854718875, 1856156927625, 194896477400625, 2900518163668125, 3698160658676859375, 1298054391195577640625, 263505041412702261046875, 122529844256906551386796875, 4043484860477916195764296875

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• frac

Comments

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

Examples

			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
		

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.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, equation 34:6:1 at page 323.

Links

• 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.
• Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1990. See p. 50.

Crossrefs

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

Programs

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 *)

Formula

a(n) = denominator((-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

Extensions

Incorrect comment by Stephen Crowley deleted by Johannes W. Meijer, Jan 19 2009