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a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (k+1)^(k-1) / (k! * (n-3*k)!).
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%I #30 Apr 24 2023 08:38:02

%S 1,1,1,-5,-23,-59,961,7351,29905,-877463,-9450719,-52724429,

%T 2282907001,31742360365,225092745697,-12992587010129,-221436656404319,

%U -1905297800257199,137972958868569025,2784953660339878507,28177036295775415561,-2459373614334806266859

%N a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * (k+1)^(k-1) / (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 - LambertW(x^3)) = LambertW(x^3)/x^3 * exp(x).

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

%Y Cf. A105785, A361916, A362523.

%K sign,easy

%O 0,4

%A _Seiichi Manyama_, Apr 24 2023