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A357334 E.g.f. satisfies A(x) = -log(1 - x) * exp(3 * A(x)).

%I A357334 #22 Feb 16 2025 08:34:04
%S A357334 0,1,7,101,2286,71064,2815812,135719352,7708432680,504204903504,
%T A357334 37327594368240,3085620116373288,281715917686701264,
%U A357334 28154794766366676888,3057177180272007475368,358397769923628731936280,45115415964514707860498688,6069465245766845367272738304
%N A357334 E.g.f. satisfies A(x) = -log(1 - x) * exp(3 * A(x)).
%H A357334 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A357334 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A357334 E.g.f.: -LambertW(3 * log(1 - x))/3.
%F A357334 a(n) = Sum_{k=1..n} (3 * k)^(k-1) * |Stirling1(n,k)|.
%F A357334 a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n - n*exp(-1)/3 - 1/2)). - _Vaclav Kotesovec_, Oct 04 2022
%F A357334 E.g.f.: Series_Reversion( 1 - exp(-x * exp(-3*x)) ). - _Seiichi Manyama_, Sep 10 2024
%t A357334 nmax = 17; A[_] = 0;
%t A357334 Do[A[x_] = -Log[1 - x]*Exp[3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A357334 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A357334 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(3*log(1-x))/3)))
%o A357334 (PARI) a(n) = sum(k=1, n, (3*k)^(k-1)*abs(stirling(n, k, 1)));
%Y A357334 Cf. A052807, A357333.
%Y A357334 Cf. A264408, A357322.
%K A357334 nonn
%O A357334 0,3
%A A357334 _Seiichi Manyama_, Sep 24 2022