A297009 Expansion of e.g.f. arcsin(x*exp(x)).

0, 1, 2, 4, 16, 104, 816, 7792, 89216, 1177920, 17603200, 294334976, 5442281472, 110221745152, 2426850793472, 57718658411520, 1474590580228096, 40274407232294912, 1171043235561185280, 36115912820342407168, 1177554628069200035840, 40471207964013864124416

Offset: 0

Author

Ilya Gutkovskiy, Dec 23 2017

Keywords

• nonn

Examples

			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! + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..100

Crossrefs

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

Programs

Maple

a:=series(arcsin(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 21; CoefficientList[Series[ArcSin[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I x Exp[x] + Sqrt[1 - x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!

PARI

first(n) = my(x='x+O('x^n)); Vec(serlaplace(asin(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

Formula

a(n) ~ sqrt(1 + LambertW(1)) * n^(n-1) / (exp(n) * LambertW(1)^n). - Vaclav Kotesovec, Mar 26 2019