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arctanh(log(x+1)-arcsinh(x))=-1/2!*x^2+3/3!*x^3-6/4!*x^4+15/5!*x^5...
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%I #7 Nov 01 2013 18:48:51

%S 0,0,-1,3,-6,15,-150,1575,-12600,101115,-1103760,14708925,-198336600,

%T 2783105325,-43768604880,754367488875,-13648397162400,262037979177975,

%U -5424089927256000,118958487086664225

%N arctanh(log(x+1)-arcsinh(x))=-1/2!*x^2+3/3!*x^3-6/4!*x^4+15/5!*x^5...

%F a(n) ~ (n-1)! * (-1)^(n+1) / 2 * (((2-exp(1))*exp(1))/(exp(1)-exp(2)+sqrt(2*(exp(1)-1)*exp(1))))^n. - _Vaclav Kotesovec_, Nov 01 2013

%t CoefficientList[Series[ArcTanh[Log[x+1]-ArcSinh[x]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 01 2013 *)

%K sign

%O 0,4

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

%E Prepended missing a(0)=0, a(1)=0 from _Vaclav Kotesovec_, Nov 01 2013