%I #14 Oct 02 2021 07:58:04
%S 0,1,2,2,-8,-76,-264,1672,36800,261648,-1443680,-66164704,-792152448,
%T 2482671424,289529373056,5294082629760,1648955815936,
%U -2474170098704128,-65494141255724544,-303927676523118080,35926135133071923200,1341060635191667045376
%N Expansion of e.g.f. arcsinh(x*exp(x)).
%H Seiichi Manyama, <a href="/A297010/b297010.txt">Table of n, a(n) for n = 0..100</a>
%e arcsinh(x*exp(x)) = x^1/1! + 2*x^2/2! + 2*x^3/3! - 8*x^4/4! - 76*x^5/5! - 264*x^6/6! + ...
%p a:=series(arcsinh(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 21; CoefficientList[Series[ArcSinh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 21; CoefficientList[Series[Log[x Exp[x] + Sqrt[1 + x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) first(n) = my(x='x+O('x^n)); Vec(serlaplace(asinh(exp(x)*x)), -n) \\ _Iain Fox_, Dec 23 2017
%Y Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297009.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Dec 23 2017