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Column 3 of A355996.
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a[n_] := n!/(6 * Floor[n/3]!); Array[a, 22, 3] (* Amiram Eldar, Jul 22 2022 *)
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a(n) = A355988(n)/6.
E.g.f.: (1 - x^3) * (exp(x^3) - 1)/( 6 * (1 - x)).