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A012027
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E.g.f. cosh(sin(arctan(x))) = cosh(x/sqrt(1+x^2)) (even powers only).
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0
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1, 1, -11, 301, -15287, 1239481, -146243459, 23567903269, -4951201340399, 1307274054385393, -420773143716828539, 160635990248839962781, -70764171306270411101351, 34822234810202848704345001
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = ((2*n)!*sum(k=0..n, C(n-1,n-k)/(2*k)!*(-1)^(n-k))). - Vladimir Kruchinin, Jun 17 2011
E.g.f.: cosh(x/sqrt(1+x^2)) = 1 + x^2/(G(0) - x^2) where G(k)= (2*k+2)*(2*k+1)*(1+x^2) + x^2 - (2*k+2)*(2*k+1)*x^2*(1+x^2)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Aug 06 2012
D-finite with recurrence: a(n) = -(12*n^2 - 24*n + 11)*a(n-1) - 12*(n-2)*(n-1)*(2*n-3)^2*a(n-2) - 16*(n-3)*(n-2)^2*(n-1)*(2*n-5)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Nov 09 2013
Lim sup n->infinity |a(n)|/(2^(2*n+2/3) * exp(3/4*(2*n)^(1/3)-2*n) * n^(2*n-1/3) / sqrt(3)) = 1. - Vaclav Kotesovec, Nov 09 2013
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EXAMPLE
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cosh(sin(arctan(x))) = 1+1/2!*x^2-11/4!*x^4+301/6!*x^6-15287/8!*x^8...
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MATHEMATICA
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Table[n!*SeriesCoefficient[Cosh[x/Sqrt[1+x^2]], {x, 0, n}], {n, 0, 40, 2}] (* Vaclav Kotesovec, Nov 08 2013 *)
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PROG
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(Maxima)
a(n):=((2*n)!*sum(binomial(n-1, n-k)/(2*k)!*(-1)^(n-k), k, 0, n)); [Vladimir Kruchinin, Jun 17 2011]
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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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STATUS
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approved
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