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A356798 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^3.

%I A356798 #23 Feb 16 2025 08:34:03
%S A356798 1,0,2,3,88,425,13476,130417,4543120,71005041,2723297860,60685651961,
%T A356798 2564091428856,75166650583609,3496499475113932,127585829832674865,
%U A356798 6521845096842043936,284745004488498858209,15950013722559213419412,809403234909367349670409
%N A356798 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^3.
%H A356798 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356798 a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
%F A356798 E.g.f.: A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
%F A356798 E.g.f.: A(x) = exp( -LambertW(3 * x * (1 - exp(x)))/3 ).
%F A356798 E.g.f.: A(x) = ( LambertW(3 * x * (1 - exp(x)))/(3 * x * (1 - exp(x))) )^(1/3).
%t A356798 nmax = 19; A[_] = 1;
%t A356798 Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
%t A356798 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A356798 (PARI) a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
%o A356798 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
%o A356798 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*(1-exp(x)))/3)))
%o A356798 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*(1-exp(x)))/(3*x*(1-exp(x))))^(1/3)))
%Y A356798 Cf. A052506, A355843, A356797.
%Y A356798 Cf. A356789.
%K A356798 nonn
%O A356798 0,3
%A A356798 _Seiichi Manyama_, Aug 28 2022