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Expansion of e.g.f. of exp(arcsinh(x)/exp(x)).
1

%I #18 Nov 20 2016 03:51:50

%S 1,1,-1,-3,9,25,-145,-539,3857,30129,-192673,-2816979,17218201,

%T 364417417,-2297788337,-63429766763,417350455329,14529736975457,

%U -99514205291841,-4252720982041379,30285000226110761

%N Expansion of e.g.f. of exp(arcsinh(x)/exp(x)).

%H G. C. Greubel, <a href="/A013572/b013572.txt">Table of n, a(n) for n = 0..400</a>

%F E.g.f.: (x+sqrt(1+x^2))^(1/exp(x)). - _Vaclav Kotesovec_, Nov 01 2013

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

%e exp(arcsinh(x)/exp(x)) = 1 + x - 1/2!*x^2 - 3/3!*x^3 + 9/4!*x^4 + 25/5!*x^5 - ..

%t With[{nn=30},CoefficientList[Series[Exp[ArcSinh[x]/Exp[x]],{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 19 2011 *)

%o (PARI) x='x + O('x^50); Vec(serlaplace(exp(asinh(x)/exp(x)))) \\ _G. C. Greubel_, Nov 20 2016

%K sign

%O 0,4

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