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A227463 E.g.f. equals the series reversion of arcsinh(x) / exp(x).

%I A227463 #20 Jul 22 2018 09:24:28
%S A227463 1,2,10,80,876,12192,206144,4104704,94092112,2440642560,70676191840,
%T A227463 2260198354944,79113937385536,3008546200346624,123513154739070976,
%U A227463 5444598073252904960,256489070938397360384,12859678961654923395072,683701585124386481758720
%N A227463 E.g.f. equals the series reversion of arcsinh(x) / exp(x).
%C A227463 Note that arcsinh(x) = log(sqrt(1+x^2) + x).
%F A227463 E.g.f. A(x) satisfies: A(x) = sinh(x*exp(A(x))).
%F A227463 a(n) ~ n^(n-1) * sqrt((1+s^2)/(1+s+s^2)) * (sqrt(1+s^2)/exp(1-s))^n, where s = 0.84184323411403778647... is the root of the equation sqrt(1+s^2)*arcsinh(s) = 1. - _Vaclav Kotesovec_, Jan 13 2014
%e A227463 E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 876*x^5/5! + 12192*x^6/6! + ...
%e A227463 where A( arcsinh(x)/exp(x) ) = x.
%t A227463 Rest[CoefficientList[InverseSeries[Series[ArcSinh[x] / Exp[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o A227463 (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(asinh(X)/exp(X)), n)}
%o A227463 for(n=1,25,print1(a(n),", "))
%o A227463 (PARI) {a(n)=local(A=x); for(i=1,n,A=sinh(x*exp(A+x*O(x^n)))); n!*polcoeff(A, n)}
%o A227463 for(n=1,25,print1(a(n),", "))
%Y A227463 Cf. A227464.
%K A227463 nonn
%O A227463 1,2
%A A227463 _Paul D. Hanna_, Jul 13 2013