A331727 E.g.f.: -LambertW(-x/(1 + x)) / (1 + x).

0, 1, -2, 9, -32, 225, -1044, 11515, -53696, 1056321, -2809700, 164953371, 374457744, 42734920657, 415505963068, 17518516958475, 310367497789696, 10529847396874497, 258747727039635132, 8599295530916762779, 258064489282796717200, 9014901067536225062481

Offset: 0

Author

Ilya Gutkovskiy, Jan 25 2020

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000169, A060356, A231797, A277511, A331726.

Programs

Mathematica

nmax = 21; CoefficientList[Series[-LambertW[-x/(1 + x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k Binomial[n, k]^2 k! (n - k)^(n - k - 1), {k, 0, n - 1}], {n, 0, 21}]

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} (-1)^k * 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