%I #26 Mar 05 2024 07:46:32
%S 1,0,0,6,36,150,1620,24486,293076,3843510,68254740,1311687366,
%T 25479935316,552545882070,13437670215060,345157499363046,
%U 9370414233900756,274413997443811830,8572526271218671380,281754864204797848326,9767868351458229261396
%N E.g.f. satisfies log(A(x)) = (exp(x) - 1)^3 * A(x).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (k+1)^(k-1) * Stirling2(n,3*k)/k!.
%F E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(3*k) / k!.
%F E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^3) ).
%F E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^3)/(exp(x) - 1)^3.
%F a(n) ~ sqrt(1 + exp(1/3)) * 3^n * n^(n-1) / (exp(n-1) * (3*log(1 + exp(1/3)) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Sep 27 2023
%t nmax = 20; A[_] = 1;
%t Do[A[x_] = Exp[(Exp[x] - 1)^3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o (PARI) a(n) = sum(k=0, n\3, (3*k)!*(k+1)^(k-1)*stirling(n, 3*k, 2)/k!);
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(3*k)/k!)))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^3))))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^3)/(exp(x)-1)^3))
%Y Cf. A052880, A357009.
%Y Cf. A353664, A357025.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Sep 09 2022