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A002430
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Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x).
(Formerly M2100 N0832)
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18
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1, 1, 2, 17, 62, 1382, 21844, 929569, 6404582, 443861162, 18888466084, 113927491862, 58870668456604, 8374643517010684, 689005380505609448, 129848163681107301953, 1736640792209901647222, 418781231495293038913922
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OFFSET
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1,3
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COMMENTS
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REFERENCES
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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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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].
Eric Weisstein's World of Mathematics, Tangent
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FORMULA
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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
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
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EXAMPLE
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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)!).
tanh(x) = x - (1/3)*x^3 + (2/15)*x^5 - (17/315)*x^7 + (62/2835)*x^9 - (1382/155925)*x^11 + ...
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MAPLE
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R := n -> (-1)^floor(n/2)*(4^n-2^n)*Zeta(1-n)/(n-1)!:
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = numerator( (-1)^(n-1)*4^n*(4^n-1)*bernfrac(2*n)/(2*n)! ); \\ G. C. Greubel, Jul 03 2019
(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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EXTENSIONS
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More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003
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STATUS
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approved
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