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%I #22 Sep 11 2024 10:05:32
%S 0,1,5,36,379,5461,100476,2250613,59432141,1807959042,62262816157,
%T 2394551966401,101724440338494,4730814590128615,239057921691911861,
%U 13042779411190737420,764136645388807739239,47846833035272035228849,3188740106752561252031364
%N Expansion of e.g.f. -LambertW((1 - exp(3*x))/3).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * Stirling2(n,k).
%F a(n) ~ 3^(n - 1/2) * sqrt(exp(1) + 3) * n^(n-1) / (exp(n) * (log(exp(1) + 3) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Oct 04 2022
%F E.g.f.: Series_Reversion( (log(1 + 3 * x * exp(-x)))/3 ). - _Seiichi Manyama_, Sep 11 2024
%t With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[3*x])/3], {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)
%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw((1-exp(3*x))/3))))
%o (PARI) a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*stirling(n, k, 2));
%Y Cf. A048802, A356000.
%Y Cf. A357336.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Sep 24 2022