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A002430 Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).
(Formerly M2100 N0832)
17

%I M2100 N0832

%S 1,1,2,17,62,1382,21844,929569,6404582,443861162,18888466084,

%T 113927491862,58870668456604,8374643517010684,689005380505609448,

%U 129848163681107301953,1736640792209901647222,418781231495293038913922

%N Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).

%C a(n) appears to be a multiple of A046990(n) (checked up to n=250). - _Ralf Stephan_, Mar 30 2004

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

%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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe and Seiichi Manyama, <a href="/A002430/b002430.txt">Table of n, a(n) for n = 1..276</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 R. W. van der Waall, <a href="http://dx.doi.org/10.1016/0022-314X(73)90050-4">On a property of tan x</a>, J. Number Theory 5 (1973) 242-244.

%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) is the numerator of (-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) = numerator(R(2*n)) and A036279(n) = denominator(R(2*n)). _Peter Luschny_, Aug 25 2015

%e tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... =

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

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

%p seq(numer(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[Numerator@ f[2 n], {n, 18}] (* _Michael De Vlieger_, Aug 25 2015 *)

%Y Cf. A036279 (denominators), A000182, A099612, A160469, A156769.

%K nonn,easy,frac

%O 1,3

%A _N. J. A. Sloane_

%E More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003

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Last modified August 20 14:38 EDT 2018. Contains 313917 sequences. (Running on oeis4.)