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%I #14 Apr 16 2023 23:03:29
%S 1,1,1,-2,-15,-49,241,3186,17473,-136835,-2591199,-19940194,214217521,
%T 5280969123,52303886545,-714177220574,-21687847310079,
%U -262685369226919,4351534043729473,157014580915662750,2248361900084617201,-43790588385118719689
%N a(n) = n! * Sum_{k=0..floor(n/3)} (-n/6)^k * binomial(n-2*k,k)/(n-2*k)!.
%H Winston de Greef, <a href="/A362303/b362303.txt">Table of n, a(n) for n = 0..441</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = n! * [x^n] exp(x - n*x^3/6).
%F E.g.f.: exp( ( 2*LambertW(x^3/2) )^(1/3) ) / (1 + LambertW(x^3/2)).
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((2*lambertw(x^3/2))^(1/3))/(1+lambertw(x^3/2))))
%Y Main diagonal of A362302.
%K sign
%O 0,4
%A _Seiichi Manyama_, Apr 15 2023