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A357333 E.g.f. satisfies A(x) = -log(1 - x) * exp(2 * A(x)).

%I A357333 #21 Feb 16 2025 08:34:04
%S A357333 0,1,5,50,778,16604,451668,14947568,582982160,26187136128,
%T A357333 1331445995520,75589772147328,4739901861071232,325353447339098112,
%U A357333 24264683011603485696,1953776475810372817920,168924939633683095452672,15609228287753846217412608
%N A357333 E.g.f. satisfies A(x) = -log(1 - x) * exp(2 * A(x)).
%H A357333 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A357333 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A357333 E.g.f.: -LambertW(2 * log(1 - x))/2.
%F A357333 a(n) = Sum_{k=1..n} (2 * k)^(k-1) * |Stirling1(n,k)|.
%F A357333 a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2)-1)^(n - 1/2) * exp(n - n*exp(-1)/2 - 1/2)). - _Vaclav Kotesovec_, Oct 04 2022
%F A357333 E.g.f.: Series_Reversion( 1 - exp(-x * exp(-2*x)) ). - _Seiichi Manyama_, Sep 09 2024
%t A357333 nmax = 17; A[_] = 0;
%t A357333 Do[A[x_] = -Log[1 - x]*Exp[2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A357333 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A357333 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(2*log(1-x))/2)))
%o A357333 (PARI) a(n) = sum(k=1, n, (2*k)^(k-1)*abs(stirling(n, k, 1)));
%Y A357333 Cf. A052807, A357334.
%Y A357333 Cf. A264407, A357321.
%K A357333 nonn
%O A357333 0,3
%A A357333 _Seiichi Manyama_, Sep 24 2022