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%I #11 Sep 25 2024 10:29:38
%S 1,0,0,6,24,0,2520,35280,141120,6048000,181440000,1995840000,
%T 51831964800,2280127449600,47882676441600,1192991325926400,
%U 59048471978496000,1942527607308288000,56983429057076121600,2842216483159788134400,126830901998902413312000
%N E.g.f. satisfies A(x) = exp(x^3 * (1 + x) * A(x)^3).
%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( -LambertW(-3*x^3 * (1+x))/3 ).
%F a(n) = n! * Sum_{k=0..floor(n/3)} (3*k+1)^(k-1) * binomial(k,n-3*k)/k!.
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x^3*(1+x))/3)))
%o (PARI) a(n) = n!*sum(k=0, n\3, (3*k+1)^(k-1)*binomial(k, n-3*k)/k!);
%Y Cf. A362771, A376492.
%Y Cf. A376477.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Sep 25 2024