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A357027 E.g.f. satisfies A(x) = 1/(1 - x)^(log(1 - x)^2 * A(x)).

%I A357027 #19 Feb 16 2025 08:34:04
%S A357027 1,0,0,6,36,210,2430,32424,426552,6575304,118916640,2328078456,
%T A357027 49421111256,1153979875152,29201577206256,791744021665344,
%U A357027 22988121190902720,712541051083100160,23447653175729566080,816434611464004145280,30009023179153182132480
%N A357027 E.g.f. satisfies A(x) = 1/(1 - x)^(log(1 - x)^2 * A(x)).
%H A357027 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A357027 E.g.f. satisfies log(A(x)) = -log(1 - x)^3 * A(x).
%F A357027 a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (k+1)^(k-1) * |Stirling1(n,3*k)|/k!.
%F A357027 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-log(1 - x))^(3*k) / k!.
%F A357027 E.g.f.: A(x) = exp( -LambertW(log(1-x)^3) ).
%F A357027 E.g.f.: A(x) = LambertW(log(1 - x)^3)/log(1 - x)^3.
%t A357027 m = 21; (* number of terms *)
%t A357027 A[_] = 0;
%t A357027 Do[A[x_] = 1/(1 - x)^(Log[1 - x]^2*A[x]) + O[x]^m // Normal, {m}];
%t A357027 CoefficientList[A[x], x]*Range[0, m - 1]! (* _Jean-François Alcover_, Sep 12 2022 *)
%o A357027 (PARI) a(n) = sum(k=0, n\3, (3*k)!*(k+1)^(k-1)*abs(stirling(n, 3*k, 1))/k!);
%o A357027 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-log(1-x))^(3*k)/k!)))
%o A357027 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(log(1-x)^3))))
%o A357027 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(log(1-x)^3)/log(1-x)^3))
%Y A357027 Cf. A052813, A357026.
%K A357027 nonn
%O A357027 0,4
%A A357027 _Seiichi Manyama_, Sep 09 2022