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A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

%I A361069 #21 Feb 16 2025 08:34:04
%S A361069 1,1,-3,40,-719,18396,-598157,23713726,-1108701519,59735988424,
%T A361069 -3644505746549,248358786667674,-18697767289462967,
%U A361069 1541202721786228060,-138046868771541971373,13351368704222195975206,-1386710317839048140282783,153939247458296219191539984
%N A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).
%H A361069 Winston de Greef, <a href="/A361069/b361069.txt">Table of n, a(n) for n = 0..338</a>
%H A361069 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A361069 a(n) = n! * Sum_{k=0..n} (-3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
%F A361069 E.g.f.: exp( LambertW(3*x/(1-x))/3 ).
%F A361069 E.g.f.: 1 / ( (1-x)/(3*x) * LambertW(3*x/(1-x)) )^(1/3).
%F A361069 a(n) ~ (-1)^(n+1) * 3^(-3/2) * exp(-1/3) * (3 - exp(-1))^(n + 1/2) * n^(n-1). - _Vaclav Kotesovec_, Apr 22 2024
%t A361069 nmax = 20; A[_] = 1;
%t A361069 Do[A[x_] = Exp[x/((1 - x)*A[x]^3)] + O[x]^(nmax+1) // Normal, {nmax}];
%t A361069 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A361069 (PARI) a(n) = n!*sum(k=0, n, (-3*k+1)^(k-1)*binomial(n-1, n-k)/k!);
%o A361069 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*x/(1-x))/3)))
%o A361069 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/((1-x)/(3*x)*lambertw(3*x/(1-x)))^(1/3)))
%Y A361069 Cf. A052868, A361065, A361066, A361067, A361068.
%K A361069 sign
%O A361069 0,3
%A A361069 _Seiichi Manyama_, Mar 01 2023