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%I #9 Mar 27 2019 03:50:51
%S 0,1,-3,9,-32,148,-853,6027,-49576,470624,-5005137,59454923,
%T -774282632,11035740844,-169997137269,2826070412955,-50256453936368,
%U 954657085889760,-19247168446169665,411277539407862707,-9269937746437524256,220085825544691181500,-5483977295221312280757
%N Expansion of e.g.f. log(1 + arcsin(x))*exp(-x).
%e log(1 + arcsin(x))*exp(-x) = x/1! - 3*x^2/2! + 9*x^3/3! - 32*x^4/4! + 148*x^5/5! - 853*x^6/6! + ...
%p a:=series(log(1+arcsin(x))*exp(-x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 22; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) x='x+O('x^99); concat([0], Vec(serlaplace(exp(-x)*log(1+asin(x))))) \\ _Altug Alkan_, Dec 28 2017
%Y Cf. A009337, A009356, A009374, A009393, A189815, A296436, A297210, A297211, A297213.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Dec 27 2017