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%I #15 Jan 26 2020 17:22:50
%S 1,1,2,5,16,69,368,2169,14208,109929,970752,8995821,88341504,
%T 988161069,12276025344,154843019169,2009594658816,29484826539345,
%U 476778061430784,7588488203093205,121001549512310784,2205431202369899925,44538441694414110720,852615914764223422665
%N Expansion of e.g.f. 1/(1 - arcsinh(x)).
%C a(48) is negative. - _Vaclav Kotesovec_, Jan 26 2020
%H Vaclav Kotesovec, <a href="/A296675/b296675.txt">Table of n, a(n) for n = 0..400</a>
%F E.g.f.: 1/(1 - log(x + sqrt(1 + x^2))).
%F a(n) ~ 8*((4 - Pi^2)*sin(Pi*n/2) - 4*Pi*cos(Pi*n/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n)). - _Vaclav Kotesovec_, Dec 18 2017
%e 1/(1 - arcsinh(x)) = 1 + x/1! + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 69*x^5/5! + ...
%p a:=series(1/(1-arcsinh(x)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # _Paolo P. Lava_, Mar 27 2019
%t nmax = 23; CoefficientList[Series[1/(1 - ArcSinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 23; CoefficientList[Series[1/(1 - Log[x + Sqrt[1 + x^2]]), {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) x='x+O('x^99); Vec(serlaplace(1/(1-log(x+sqrt(1+x^2))))) \\ _Altug Alkan_, Dec 18 2017
%Y Cf. A000111, A001818, A006154, A189780.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Dec 18 2017
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