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A356913 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).

%I A356913 #19 Feb 16 2025 08:34:03
%S A356913 1,0,0,0,4,10,40,210,784,5040,40200,369600,5285280,72072000,
%T A356913 1006889520,14760345600,210510263040,3131345817600,49229619129600,
%U A356913 818940523564800,15054020163619200,301204611031564800,6455999452413772800,146587705490513548800
%N A356913 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).
%H A356913 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356913 a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
%F A356913 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
%F A356913 E.g.f.: A(x) = exp( LambertW(-x^3/6 * log(1-x)) ).
%F A356913 E.g.f.: A(x) = -x^3/6 * log(1-x)/LambertW(-x^3/6 * log(1-x)).
%t A356913 nmax = 23; A[_] = 1;
%t A356913 Do[A[x_] = ((1 - x)^(-x^3/6))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356913 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356913 (PARI) a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));
%o A356913 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3/6*log(1-x))^k/k!)))
%o A356913 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3/6*log(1-x)))))
%o A356913 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3/6*log(1-x)/lambertw(-x^3/6*log(1-x))))
%Y A356913 Cf. A356905, A356912.
%Y A356913 Cf. A351493, A356753.
%K A356913 nonn
%O A356913 0,5
%A A356913 _Seiichi Manyama_, Sep 03 2022