A385501 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arctanh(x)) ).

1, 1, 3, 18, 165, 2040, 31815, 599760, 13268745, 337115520, 9674678475, 309554784000, 10927053262125, 421849524096000, 17682153623909775, 799730490214656000, 38820939579369572625, 2013202580708487168000, 111081054630965602057875, 6497703571257963896832000

Offset: 0

Author

Seiichi Manyama, Jul 01 2025

Keywords

• nonn

Crossrefs

Cf. A385500, A385502.

Cf. A001147, A111594, A138020, A227466, A385426.

Programs

Mathematica

nmax=20; CoefficientList[(1/x) *InverseSeries[Series[x * Exp[-ArcTanh[x]],{x,0,nmax}],x] ,x]Range[0,nmax-1]! (* Stefano Spezia, Jul 01 2025 *)

PARI

a(n) = n!*sum(k=0, n, binomial(n, k)*binomial(n/2+k+1/2, n)/(n+2*k+1));

Formula

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

E.g.f. A(x) satisfies A(x) = sqrt( (1+x*A(x))/(1-x*A(x)) ).

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

a(n) = n!/2^n * A138020(n) = n! * Sum_{k=0..n} binomial(n,k) * binomial(n/2+k+1/2,n)/(n+2*k+1).