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a(n) = n! * Sum_{k=0..floor(n/3)} (-k/6)^k / (k! * (n-3*k)!).
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%I #13 Apr 18 2023 08:28:17

%S 1,1,1,0,-3,-9,21,246,1065,-4283,-67319,-397484,2315941,45914155,

%T 343743037,-2623221054,-62980998639,-571382718039,5391435590545,

%U 152175023203432,1622112809355661,-18232162910685569,-591788241447761819,-7247966654986009490

%N a(n) = n! * Sum_{k=0..floor(n/3)} (-k/6)^k / (k! * (n-3*k)!).

%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) / (1 + LambertW(x^3/6)).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(x^3/6))))

%Y Cf. A069856, A362340, A362342.

%Y Cf. A351929, A362303, A362348.

%K sign

%O 0,5

%A _Seiichi Manyama_, Apr 17 2023