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A012025
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E.g.f. arcsinh(sin(arctan(x))) = arcsinh(x/(1+x^2)^(1/2)) (odd powers only).
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0
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1, -4, 84, -4320, 418320, -66225600, 15657364800, -5187108326400, 2296766568096000, -1310785979158656000, 937056917610253440000, -820081468493365478400000, 862301491174096979765760000
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. arcsinh(sin(arctan(x))) = arcsinh(x/(1+x^2)^(1/2)).
arcsinh(x/(1+x^2)^(1/2)) = x/sqrt(1+x^2)*(1 - x^2/(G(0)+x^2)) where G(k) = 4*k^2 + k*(10+6*x^2) + 5*x^2 + 6 + 2*x^2*(1+x^2)*(k+1)*(2*k+3)^3/G(k+1) ; (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 08 2012
a(n) ~ (-1)^(n+1) * 2^(3*n-1) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Oct 30 2013
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EXAMPLE
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arcsinh(sin(arctan(x)))=x-4/3!*x^3+84/5!*x^5-4320/7!*x^7+418320/9!*x^9...
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MATHEMATICA
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Table[n!*SeriesCoefficient[ArcSinh[x/(1+x^2)^(1/2)], {x, 0, n}], {n, 1, 40, 2}] (* Vaclav Kotesovec, Oct 30 2013 *)
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CROSSREFS
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KEYWORD
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sign
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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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STATUS
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approved
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