This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385501 #19 Jul 01 2025 09:55:52
%S A385501 1,1,3,18,165,2040,31815,599760,13268745,337115520,9674678475,
%T A385501 309554784000,10927053262125,421849524096000,17682153623909775,
%U A385501 799730490214656000,38820939579369572625,2013202580708487168000,111081054630965602057875,6497703571257963896832000
%N A385501 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arctanh(x)) ).
%F A385501 E.g.f. A(x) satisfies A(x) = exp( arctanh(x*A(x)) ).
%F A385501 E.g.f. A(x) satisfies A(x) = sqrt( (1+x*A(x))/(1-x*A(x)) ).
%F A385501 a(n) = Sum_{k=0..n} (n+1)^(k-1) * A111594(n,k).
%F A385501 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).
%t A385501 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 *)
%o A385501 (PARI) a(n) = n!*sum(k=0, n, binomial(n, k)*binomial(n/2+k+1/2, n)/(n+2*k+1));
%Y A385501 Cf. A385500, A385502.
%Y A385501 Cf. A001147, A111594, A138020, A227466, A385426.
%K A385501 nonn
%O A385501 0,3
%A A385501 _Seiichi Manyama_, Jul 01 2025