A347993 a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.

1, 2, 15, 136, 1645, 24336, 426979, 8658560, 199234809, 5128019200, 145969492471, 4552809182208, 154404454932325, 5656950010320896, 222655633595044875, 9369696305273798656, 419790650812640438641, 19950175280765680680960, 1002394352017754098219999, 53092232229227200348160000

Offset: 1

Author

Ilya Gutkovskiy, Sep 23 2021

Keywords

• nonn

Crossrefs

Cf. A000169, A000312, A001865, A063169, A133297, A134095, A277458, A347994.

Programs

Mathematica

Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest

PARI

a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ Michel Marcus, Sep 23 2021

Formula

E.g.f.: -LambertW(-x) / (1 - LambertW(-x)^2).

a(n) = n * A133297(n).