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A101927
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E.g.f. of sin(arcsinh(x)) (odd powers only).
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4
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1, -2, 20, -520, 26000, -2132000, 260104000, -44217680000, 9993195680000, -2898026747200000, 1049085682486400000, -463695871658988800000, 245758811979264064000000, -153845016299019304064000000
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OFFSET
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1,2
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COMMENTS
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Absolute values are expansion of sinh(arcsin(x)).
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LINKS
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FORMULA
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E.g.f.: sin(arcsinh(x)) = x*sqrt(1+x^2)*(1 - 5*x^2/(G(0) + 5*x^2))); G(k) = (2*k+2)*(2*k+3) - x^2*(4*k^2+8*k+5) + x^2*(2*k+2)*(2*k+3)*(4*k^2+16*k+17)/G(k+1);
for sinh(arcsin(x)) = x*sqrt(1-x^2)*(1 + 5*x^2/(G(0) - 5*x^2))); G(k) = (2*k+2)*(2*k+3) + x^2*(4*k^2+8*k+5) - x^2*(2*k+2)*(2*k+3)*(4*k^2+16*k+17)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 19 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 + (4*k^2+4*k+2)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) ~ (-1)^(n+1) * cosh(Pi/2) * 2^(2*n-1) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Oct 23 2013
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EXAMPLE
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sin(arcsinh(x)) = x - 2x^3/3! + 20x^5/5! - 520x^7/7! + 26000x^9/9! - ...
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MAPLE
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seq(coeff(series(factorial(n)*sin(arcsinh(x)), x, n+1), x, n), n=1..30, 2); # Muniru A Asiru, Jul 22 2018
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MATHEMATICA
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Table[n!*SeriesCoefficient[Sin[ArcSinh[x]], {x, 0, n}], {n, 1, 40, 2}] (* Vaclav Kotesovec, Oct 23 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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EXTENSIONS
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STATUS
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approved
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