A347994 a(n) = n! * Sum_{k=1..n-1} (-1)^(k+1) * n^(n-k-2) / (n-k-1)!.

0, 1, 4, 30, 296, 3720, 56652, 1014832, 20909520, 487198080, 12667470740, 363607605504, 11420819358456, 389646915374080, 14349217119054300, 567315485527234560, 23967624180805666208, 1077568488585047605248, 51369752823292604784420, 2588268388538639982592000

Offset: 1

Author

Ilya Gutkovskiy, Sep 23 2021

Keywords

• nonn

Crossrefs

Cf. A000169, A000435, A133297, A134095, A347993.

Programs

Mathematica

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

PARI

a(n) = n! * sum(k=1, n-1, (-1)^(k+1)*n^(n-k-2)/(n-k-1)!); \\ Michel Marcus, Sep 23 2021

Formula

E.g.f.: -LambertW(-x) - log(1 - LambertW(-x)).

a(n) = A134095(n) / n.