%I #15 Oct 01 2021 21:54:45

%S 0,1,2,4,16,104,816,7792,89216,1177920,17603200,294334976,5442281472,

%T 110221745152,2426850793472,57718658411520,1474590580228096,

%U 40274407232294912,1171043235561185280,36115912820342407168,1177554628069200035840,40471207964013864124416

%N Expansion of e.g.f. arcsin(x*exp(x)).

%H Seiichi Manyama, <a href="/A297009/b297009.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) ~ sqrt(1 + LambertW(1)) * n^(n-1) / (exp(n) * LambertW(1)^n). - _Vaclav Kotesovec_, Mar 26 2019

%e arcsin(x*exp(x)) = x^1/1! + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 104*x^5/5! + 816*x^6/6! + ...

%p a:=series(arcsin(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[ArcSin[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 21; CoefficientList[Series[-I Log[I 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(asin(exp(x)*x)), -n) \\ _Iain Fox_, Dec 23 2017

%Y Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297010.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 23 2017