%I #13 Oct 10 2024 15:11:16
%S 1,1,3,16,149,1656,22567,372184,7141689,156630448,3871782251,
%T 106504501104,3227742350197,106879926110296,3839600650843791,
%U 148746681984864856,6181806007303273073,274355581868776940256,12951023558423725477459,647956009961120527442272
%N E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x^3)).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-3*k-1) * binomial(n-2*k-1,k)/(n-3*k)!.
%F E.g.f.: exp( -LambertW(-x/(1-x^3)) ).
%F From _Vaclav Kotesovec_, Oct 10 2024: (Start)
%F E.g.f.: -LambertW(-x/(1-x^3))*(1-x^3)/x.
%F a(n) ~ sqrt(2^(2/3) * 3^(5/3) / ((2*(9 + sqrt(81 + 12*exp(3))))^(1/3) - 2*exp(1)*(3/(9 + sqrt(81 + 12*exp(3))))^(1/3)) - 2*exp(1)) * 2^(2*n/3) * 3^(4*n/3) * ((9 + sqrt(81 + 12*exp(3)))^(1/3) / (2^(1/3) * (3*(9 + sqrt(81 + 12*exp(3))))^(2/3) - 6*exp(1)))^n * n^(n-1) / exp(n - 1/2). (End)
%o (PARI) a(n) = n!*sum(k=0, n\3, (n-3*k+1)^(n-3*k-1)*binomial(n-2*k-1, k)/(n-3*k)!);
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x^3)))))
%Y Cf. A052868, A376575.
%Y Cf. A293493.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Sep 28 2024