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%I #29 Apr 16 2023 23:03:33
%S 1,1,1,4,17,51,481,3676,18369,272917,3011201,21058236,427112401,
%T 6160655359,55380250017,1423658493076,25361574327041,278603741558601,
%U 8673295084155649,183914415577719892,2387417408385462801,87273239189497636171,2146479566819857007201
%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="/A362173/b362173.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 A362043.
%Y Cf. A277614, A362300, A362301.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Apr 14 2023