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A356753 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).

%I A356753 #27 Feb 16 2025 08:34:03
%S A356753 1,0,0,0,4,10,40,210,3024,25200,225000,2217600,29974560,400720320,
%T A356753 5558957040,81340459200,1344965825280,23566775232000,432681781459200,
%U A356753 8309927446329600,170258024427580800,3679448236206220800,83235946152090547200,1962840630226968307200
%N A356753 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).
%H A356753 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356753 a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
%F A356753 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
%F A356753 E.g.f.: A(x) = exp( -LambertW(x^3/6 * log(1-x)) ).
%F A356753 E.g.f.: A(x) = LambertW(x^3/6 * log(1-x))/(x^3/6 * log(1-x)).
%t A356753 nmax = 23; A[_] = 1;
%t A356753 Do[A[x_] = 1/(1 - x)^(x^3/6*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356753 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356753 (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 A356753 (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 A356753 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*log(1-x)))))
%o A356753 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*log(1-x))/(x^3/6*log(1-x))))
%Y A356753 Cf. A351493, A356913.
%Y A356753 Cf. A355842, A356752.
%K A356753 nonn
%O A356753 0,5
%A A356753 _Seiichi Manyama_, Sep 03 2022