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%I #8 Sep 24 2021 02:52:52
%S 1,2,15,136,1645,24336,426979,8658560,199234809,5128019200,
%T 145969492471,4552809182208,154404454932325,5656950010320896,
%U 222655633595044875,9369696305273798656,419790650812640438641,19950175280765680680960,1002394352017754098219999,53092232229227200348160000
%N a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.
%F E.g.f.: -LambertW(-x) / (1 - LambertW(-x)^2).
%F a(n) = n * A133297(n).
%t Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
%t nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest
%o (PARI) a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ _Michel Marcus_, Sep 23 2021
%Y Cf. A000169, A000312, A001865, A063169, A133297, A134095, A277458, A347994.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Sep 23 2021
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