A385500 Expansion of e.g.f. exp( -LambertW(-arctanh(x)) ).

1, 1, 3, 18, 149, 1640, 22359, 366128, 6998697, 153191808, 3779353515, 103800229632, 3141633970749, 103904351855616, 3728602377979647, 144297781732300800, 5991021498320041809, 265636734347975688192, 12527923794824003280723, 626224876080360687599616

Offset: 0

Author

Seiichi Manyama, Jul 01 2025

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A385501, A385502.

Cf. A111594, A385425.

Programs

Mathematica

nmax=19; CoefficientList[Series[Exp[ -LambertW[-ArcTanh[x]]],{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Jul 01 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-atanh(x)))))

Formula

E.g.f. A(x) satisfies A(x) = exp( arctanh(x) * A(x) ).

E.g.f. A(x) satisfies A(x) = ( (1+x)/(1-x) )^(A(x)/2).

a(n) = Sum_{k=0..n} (k+1)^(k-1) * A111594(n,k).

a(n) ~ sqrt(exp(4*exp(-1)) - 1) * n^(n-1) / (2*exp(n - 3/2 + exp(-1)) * tanh(exp(-1))^n). - Vaclav Kotesovec, Jul 01 2025