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A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

%I A361066 #21 Feb 16 2025 08:34:04
%S A361066 1,1,9,148,3673,123276,5234599,269262022,16279709793,1131627199816,
%T A361066 88926737901031,7796168316687906,754414052156289265,
%U A361066 79872584117422215484,9184299004593618881655,1139822558262829096519726,151857077047173825979147969
%N A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).
%H A361066 Winston de Greef, <a href="/A361066/b361066.txt">Table of n, a(n) for n = 0..327</a>
%H A361066 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A361066 a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
%F A361066 E.g.f.: exp( -LambertW(-3*x/(1-x))/3 ).
%F A361066 E.g.f.: ( -(1-x)/(3*x) * LambertW(-3*x/(1-x)) )^(1/3).
%F A361066 a(n) ~ (1 + 3*exp(1))^(n + 1/2) * n^(n-1) / (3^(3/2) * exp(n + 1/6)). - _Vaclav Kotesovec_, Mar 02 2023
%t A361066 nmax = 20; A[_] = 1;
%t A361066 Do[A[x_] = Exp[(x/(1 - x))*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
%t A361066 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A361066 (PARI) a(n) = n!*sum(k=0, n, (3*k+1)^(k-1)*binomial(n-1, n-k)/k!);
%o A361066 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x/(1-x))/3)))
%o A361066 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-(1-x)/(3*x)*lambertw(-3*x/(1-x)))^(1/3)))
%Y A361066 Cf. A052868, A361065, A361067, A361068, A361069.
%K A361066 nonn
%O A361066 0,3
%A A361066 _Seiichi Manyama_, Mar 01 2023