A012025 - OEIS

%I #19 Oct 30 2013 16:55:15

%S 1,-4,84,-4320,418320,-66225600,15657364800,-5187108326400,

%T 2296766568096000,-1310785979158656000,937056917610253440000,

%U -820081468493365478400000,862301491174096979765760000

%N E.g.f. arcsinh(sin(arctan(x))) = arcsinh(x/(1+x^2)^(1/2)) (odd powers only).

%F E.g.f. arcsinh(sin(arctan(x))) = arcsinh(x/(1+x^2)^(1/2)).

%F 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

%F a(n) ~ (-1)^(n+1) * 2^(3*n-1) * n^(2*n-2) / exp(2*n). - _Vaclav Kotesovec_, Oct 30 2013

%e arcsinh(sin(arctan(x)))=x-4/3!*x^3+84/5!*x^5-4320/7!*x^7+418320/9!*x^9...

%t Table[n!*SeriesCoefficient[ArcSinh[x/(1+x^2)^(1/2)],{x,0,n}],{n,1,40,2}] (* _Vaclav Kotesovec_, Oct 30 2013 *)

%K sign

%O 1,2

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

%E Confirmed by _N. J. A. Sloane_, Dec 17 2011