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A362478 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).

%I A362478 #21 Feb 16 2025 08:34:05
%S A362478 1,1,1,3,33,321,2841,31641,498849,8979489,167510961,3427780401,
%T A362478 80374833441,2089382321313,58020408889353,1721768971537161,
%U A362478 55150870311938241,1897482353016075201,69322763655015214689,2676706914491568918369
%N A362478 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).
%H A362478 Seiichi Manyama, <a href="/A362478/b362478.txt">Table of n, a(n) for n = 0..399</a>
%H A362478 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A362478 E.g.f.: exp(x - LambertW(-x^3 * exp(3*x))/3) = ( -LambertW(-x^3 * exp(3*x))/x^3 )^(1/3).
%F A362478 a(n) = n! * Sum_{k=0..floor(n/3)} (1/3)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).
%t A362478 nmax = 20; A[_] = 1;
%t A362478 Do[A[x_] = Exp[x + x^3/3*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
%t A362478 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A362478 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3*exp(3*x))/3)))
%Y A362478 Column k=2 of A362490.
%Y A362478 Cf. A362474, A362491.
%Y A362478 Cf. A362390.
%K A362478 nonn
%O A362478 0,4
%A A362478 _Seiichi Manyama_, Apr 21 2023