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A012393
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E.g.f. arctanh(tan(x)*tan(x)) (even powers only).
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5
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0, 2, 16, 512, 34816, 4063232, 724566016, 183240753152, 62382319599616, 27507470234550272, 15250953398036463616, 10384044045105304174592, 8517992937742473694806016, 8285310016381680852100186112
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OFFSET
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0,2
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COMMENTS
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a(n) = 2*A000828(2*n-1). - corrected by Vaclav Kotesovec, Feb 08 2015
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LINKS
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Table of n, a(n) for n=0..13.
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FORMULA
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E.g.f.: log(sec(2*x))/2, cf. A000182. - Vladeta Jovovic, Jun 04 2005
a(n) = (-1)^(n(2n+1))4^(2n+1)(E{2n+1}(1/2)+E{2n+1}(1)); E_{n}(x) Euler polynomial. - Peter Luschny, Nov 25 2010
G.f.: 2*x/G(0) where G(k) = 1 - (2*k+2)*(2*k+4)*x/G(k+1)) (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 25 2012
G.f.: (2/G(0) - 1)*sqrt(-x), where G(k) = 1 + 1/(1 - 1/(1 + 1/(4*sqrt(-x)*(k+1)) - 1/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: 2*x*T(0), where T(k) = 1 - x*(2*k+2)*(2*k+4)/(x*(2*k+2)*(2*k+4) - 1/T(k+1) ) (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013
a(n) ~ 2^(4*n) * (2*n-1)! / Pi^(2*n). - Vaclav Kotesovec, Feb 08 2015
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EXAMPLE
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arctanh(tan(x)*tan(x)) = (2/2!)*x^2 + (16/4!)*x^4 + (512/6!)*x^6 + (34816/8!)*x^8 + ...
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MAPLE
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A012393 := n -> (-1)^(n*(2*n+1))*4^(2*n+1)*(euler(2*n+1, 1/2) + euler(2*n+1, 1)) end; # Peter Luschny, Nov 25 2010
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MATHEMATICA
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nn = 20; Table[(CoefficientList[Series[ArcTanh[Tan[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 08 2015 *)
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CROSSREFS
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Sequence in context: A140311 A012389 A009710 * A189899 A189335 A189773
Adjacent sequences: A012390 A012391 A012392 * A012394 A012395 A012396
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KEYWORD
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nonn
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AUTHOR
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Patrick Demichel (patrick.demichel(AT)hp.com)
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EXTENSIONS
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a(0)=0 prepended by Joerg Arndt, Oct 26 2012
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STATUS
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approved
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