A331726 E.g.f.: -LambertW(-x/(1 - x)) / (1 - x).

0, 1, 6, 45, 448, 5825, 95796, 1926043, 45944256, 1269187137, 39840825700, 1400286658331, 54462564354672, 2321934762267601, 107664031299459012, 5393893268767761675, 290341440380472614656, 16710435419661861992705, 1024009456958258244673860

Offset: 0

Author

Ilya Gutkovskiy, Jan 25 2020

Keywords

• nonn

Crossrefs

Cf. A000169, A052871, A245496, A277505, A331727.

Programs

Mathematica

nmax = 18; CoefficientList[Series[-LambertW[-x/(1 - x)]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! (n - k)^(n - k - 1), {k, 0, n - 1}], {n, 0, 18}]

PARI

seq(n)={Vec(serlaplace(-lambertw(-x/(1 - x) + O(x*x^n)) / (1 - x)), -(n+1))} \\ Andrew Howroyd, Jan 25 2020

Formula

a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * k! * (n - k)^(n - k - 1).

a(n) ~ (1 + exp(-1))^(n + 3/2) * n^(n-1). - Vaclav Kotesovec, Jan 26 2020