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A356911 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).

%I A356911 #16 Feb 16 2025 08:34:03
%S A356911 1,0,0,0,24,60,240,1260,-12096,-120960,-1144800,-11642400,190270080,
%T A356911 4670265600,81378198720,1348668921600,-880532674560,-406217626214400,
%U A356911 -13255586359142400,-343166884178227200,-3137937973466572800,72862796986940620800
%N A356911 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).
%H A356911 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356911 a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.
%F A356911 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3 * log(1-x))^k / k!.
%F A356911 E.g.f.: A(x) = exp( LambertW(-x^3 * log(1-x)) ).
%F A356911 E.g.f.: A(x) = -x^3 * log(1-x)/LambertW(-x^3 * log(1-x)).
%t A356911 nmax = 21; A[_] = 1;
%t A356911 Do[A[x_] = ((1 - x)^(-x^3))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356911 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356911 (PARI) a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
%o A356911 (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 A356911 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3*log(1-x)))))
%o A356911 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3*log(1-x)/lambertw(-x^3*log(1-x))))
%Y A356911 Cf. A356905, A356910.
%Y A356911 Cf. A353229, A356099.
%K A356911 sign
%O A356911 0,5
%A A356911 _Seiichi Manyama_, Sep 03 2022