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A356099 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3 * A(x)).

%I A356099 #21 Feb 16 2025 08:34:03
%S A356099 1,0,0,0,24,60,240,1260,68544,604800,5508000,54885600,1877420160,
%T A356099 32069157120,499522645440,7832035411200,236207887534080,
%U A356099 5868136834560000,133085307920947200,2941187195765145600,91568561750088652800,2857211689810118860800
%N A356099 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3 * A(x)).
%H A356099 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356099 a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.
%F A356099 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^3 * log(1-x))^k / k!.
%F A356099 E.g.f.: A(x) = exp( -LambertW(x^3 * log(1-x)) ).
%F A356099 E.g.f.: A(x) = LambertW(x^3 * log(1-x))/(x^3 * log(1-x)).
%t A356099 nmax = 21; A[_] = 1;
%t A356099 Do[A[x_] = 1/(1 - x)^(x^3*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356099 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356099 (PARI) a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
%o A356099 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^3*log(1-x))^k/k!)))
%o A356099 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3*log(1-x)))))
%o A356099 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3*log(1-x))/(x^3*log(1-x))))
%Y A356099 Cf. A353229, A356911.
%K A356099 nonn
%O A356099 0,5
%A A356099 _Seiichi Manyama_, Sep 03 2022