Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #20 Mar 04 2024 08:47:27
%S 1,1,1,3,33,321,2841,31641,498849,8979489,167510961,3427780401,
%T 80374833441,2089382321313,58020408889353,1721768971537161,
%U 55150870311938241,1897482353016075201,69322763655015214689,2676706914491568918369
%N E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).
%H Seiichi Manyama, <a href="/A362478/b362478.txt">Table of n, a(n) for n = 0..399</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: exp(x - LambertW(-x^3 * exp(3*x))/3) = ( -LambertW(-x^3 * exp(3*x))/x^3 )^(1/3).
%F a(n) = n! * Sum_{k=0..floor(n/3)} (1/3)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).
%t nmax = 20; A[_] = 1;
%t Do[A[x_] = Exp[x + x^3/3*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3*exp(3*x))/3)))
%Y Column k=2 of A362490.
%Y Cf. A362474, A362491.
%Y Cf. A362390.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Apr 21 2023