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%I #14 Sep 24 2022 08:15:18
%S 0,0,0,0,4,10,40,210,2464,20160,178800,1755600,22323840,289729440,
%T 3950069760,57127870800,921032555520,15786602832000,284810759251200,
%U 5394363163862400,108742028591923200,2312415679065811200,51543520889668684800,1199641884471310156800
%N Expansion of e.g.f. -LambertW(x^3/6 * log(1-x)).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = n! * Sum_{k=1..floor(n/4)} k^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
%t With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3/6 * Log[1 - x]], {x, 0, m}], x]] (* _Amiram Eldar_, Sep 24 2022 *)
%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3/6*log(1-x)))))
%o (PARI) a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));
%Y Cf. A052807, A355994, A357265.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Sep 24 2022