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E.g.f. satisfies A(x) = x * exp(x - 3 * A(x)).
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%I #9 Feb 09 2023 09:38:17

%S 0,1,-4,48,-896,22880,-743232,29337280,-1363752448,72979407360,

%T -4419108684800,298730433250304,-22300928914403328,

%U 1822195561572585472,-161756111552270491648,15501595224386724126720,-1595092357302221461127168,175405731698165304882495488

%N E.g.f. satisfies A(x) = x * exp(x - 3 * A(x)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: A(x) = LambertW(3*x * exp(x))/3.

%F a(n) = Sum_{k=1..n} (-3)^(k-1) * k^(n-1) * binomial(n,k).

%o (PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(lambertw(3*x*exp(x))/3)))

%o (PARI) a(n) = sum(k=1, n, (-3)^(k-1)*k^(n-1)*binomial(n, k));

%Y Cf. A216857, A360481, A360482, A360483.

%K sign

%O 0,3

%A _Seiichi Manyama_, Feb 09 2023