%I #13 Mar 27 2019 10:04:55

%S 0,1,2,5,20,109,654,4353,34824,324441,3244410,34795485,417545820,

%T 5536151685,77506123590,1144330385625,18309286170000,315366695240625,

%U 5676600514331250,106667957800963125,2133359156019262500,45229212438054868125,995042673637207098750,22696937952367956440625

%N Expansion of e.g.f. arcsinh(x)/(1 - x).

%F E.g.f.: log(x + sqrt(1 + x^2))/(1 - x).

%F a(n) ~ n! * log(1 + sqrt(2)). - _Vaclav Kotesovec_, Dec 20 2017

%e arcsinh(x)/(1 - x) = x/1! + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + ...

%p a:=series(arcsinh(x)/(1 - x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # _Paolo P. Lava_, Mar 27 2019

%t nmax = 23; CoefficientList[Series[ArcSinh[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 23; CoefficientList[Series[Log[x + Sqrt[1 + x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

%o (PARI) first(n) = x='x+O('x^n); Vec(serlaplace(asinh(x)/(1 - x)), -n) \\ _Iain Fox_, Dec 19 2017

%Y Cf. A001818, A009628, A081358, A186763, A281964, A296726.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 19 2017