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A357338 E.g.f. satisfies A(x) = log(1 + x) * exp(3 * A(x)).

%I A357338 #26 Feb 16 2025 08:34:04
%S A357338 0,1,5,65,1302,35904,1260372,53796168,2704942440,156602951568,
%T A357338 10260496538640,750563024381928,60636437884772208,5362045857366832152,
%U A357338 515154874732515894744,53432840588453561773080,5950904875941534263739648,708296073287989866587094528
%N A357338 E.g.f. satisfies A(x) = log(1 + x) * exp(3 * A(x)).
%H A357338 Seiichi Manyama, <a href="/A357338/b357338.txt">Table of n, a(n) for n = 0..336</a>
%H A357338 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H A357338 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A357338 E.g.f.: -LambertW(-3 * log(1 + x))/3.
%F A357338 a(n) = Sum_{k=1..n} (3 * k)^(k-1) * Stirling1(n,k).
%F A357338 a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n - 1/2 + exp(-1)/6)). - _Vaclav Kotesovec_, Oct 04 2022
%F A357338 E.g.f.: Series_Reversion( exp(x * exp(-3*x)) - 1 ). - _Seiichi Manyama_, Sep 10 2024
%t A357338 nmax = 20; A[_] = 0;
%t A357338 Do[A[x_] = Log[1 + x]*Exp[3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A357338 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A357338 (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-3*log(1+x))/3)))
%o A357338 (PARI) a(n) = sum(k=1, n, (3*k)^(k-1)*stirling(n, k, 1));
%Y A357338 Cf. A277489, A357337.
%Y A357338 Cf. A349505.
%K A357338 nonn
%O A357338 0,3
%A A357338 _Seiichi Manyama_, Sep 24 2022