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%I #9 Jan 26 2020 09:07:07
%S 0,1,-2,9,-32,225,-1044,11515,-53696,1056321,-2809700,164953371,
%T 374457744,42734920657,415505963068,17518516958475,310367497789696,
%U 10529847396874497,258747727039635132,8599295530916762779,258064489282796717200,9014901067536225062481
%N E.g.f.: -LambertW(-x/(1 + x)) / (1 + x).
%H Vaclav Kotesovec, <a href="/A331727/b331727.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = Sum_{k=0..n-1} (-1)^k * binomial(n,k)^2 * k! * (n - k)^(n - k - 1).
%F a(n) ~ (1 - exp(-1))^(n + 3/2) * n^(n-1). - _Vaclav Kotesovec_, Jan 26 2020
%t nmax = 21; CoefficientList[Series[-LambertW[-x/(1 + x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^k Binomial[n, k]^2 k! (n - k)^(n - k - 1), {k, 0, n - 1}], {n, 0, 21}]
%o (PARI) seq(n)={Vec(serlaplace(-lambertw(-x/(1 + x) + O(x*x^n)) / (1 + x)), -(n+1))} \\ _Andrew Howroyd_, Jan 25 2020
%Y Cf. A000169, A060356, A231797, A277511, A331726.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jan 25 2020
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