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sech(arcsin(x)*arcsin(x))=1-12/4!*x^4-240/6!*x^6-1008/8!*x^8...
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%I #7 Feb 08 2015 05:28:33

%S 1,0,-12,-240,-1008,378240,32163648,-363487488,-690754821888,

%T -145507353366528,-11585562008128512,3150884675987558400,

%U 573373842109571665920,-1149448245885758287872000

%N sech(arcsin(x)*arcsin(x))=1-12/4!*x^4-240/6!*x^6-1008/8!*x^8...

%F a(n) ~ -sqrt(Pi) * sinh(Pi^2/4) * (2*n)! / (n^(3/2) * (1 + cosh(Pi^2/2))). - _Vaclav Kotesovec_, Feb 08 2015

%t nn = 20; Table[(CoefficientList[Series[Sech[ArcSin[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* _Vaclav Kotesovec_, Feb 08 2015 *)

%K sign

%O 0,3

%A Patrick Demichel (patrick.demichel(AT)hp.com)