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A002430 Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).

%I A002430 M2100 N0832 #64 Feb 16 2025 08:32:25
%S A002430 1,1,2,17,62,1382,21844,929569,6404582,443861162,18888466084,
%T A002430 113927491862,58870668456604,8374643517010684,689005380505609448,
%U A002430 129848163681107301953,1736640792209901647222,418781231495293038913922
%N A002430 Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).
%C A002430 a(n) appears to be a multiple of A046990(n) (checked up to n=250). - _Ralf Stephan_, Mar 30 2004
%C A002430 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 A002430 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%D A002430 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A002430 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 A002430 H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 329.
%D A002430 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002430 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002430 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 A002430 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 A002430 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 A002430 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 A002430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperbolicTangent.html">Hyperbolic Tangent</a>.
%H A002430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Tangent.html">Tangent</a>.
%H A002430 Herbert S. Wilf, <a href="https://www2.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, Academic Press, NY, 1990. See p. 50.
%F A002430 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 A002430 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 A002430 tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + ... =
%e A002430   x + (1/3)*x^3 + (2/15)*x^5 + (17/315)*x^7 + (62/2835)*x^9 + ... =
%e A002430   Sum_{n >= 1} (2^(2n) - 1) * (2x)^(2n-1) * |bernoulli_2n| / (n*(2n-1)!).
%e A002430 tanh(x) = x - (1/3)*x^3 + (2/15)*x^5 - (17/315)*x^7 + (62/2835)*x^9 - (1382/155925)*x^11 + ...
%p A002430 R := n -> (-1)^floor(n/2)*(4^n-2^n)*Zeta(1-n)/(n-1)!:
%p A002430 seq(numer(R(2*n)), n=1..20); # _Peter Luschny_, Aug 25 2015
%t A002430 a[n_]:= (-1)^Floor[n/2]*(4^n - 2^n)*Zeta[1-n]/(n-1)!; Table[Numerator@ a[2n], {n, 20}] (* _Michael De Vlieger_, Aug 25 2015 *)
%o A002430 (PARI) a(n) = numerator( (-1)^(n-1)*4^n*(4^n-1)*bernfrac(2*n)/(2*n)! ); \\ _G. C. Greubel_, Jul 03 2019
%o A002430 (Magma) [Numerator( (-1)^(n-1)*4^n*(4^n-1)*Bernoulli(2*n)/Factorial(2*n) ): n in [1..20]]; // _G. C. Greubel_, Jul 03 2019
%o A002430 (Sage) [numerator( (-1)^(n-1)*4^n*(4^n-1)*bernoulli(2*n)/factorial(2*n)  ) for n in (1..20)] # _G. C. Greubel_, Jul 03 2019
%Y A002430 Cf. A036279 (denominators), A000182, A099612, A160469, A156769.
%K A002430 nonn,easy,frac
%O A002430 1,3
%A A002430 _N. J. A. Sloane_
%E A002430 More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003