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%I #9 Jan 26 2020 09:01:17
%S 0,1,6,45,448,5825,95796,1926043,45944256,1269187137,39840825700,
%T 1400286658331,54462564354672,2321934762267601,107664031299459012,
%U 5393893268767761675,290341440380472614656,16710435419661861992705,1024009456958258244673860
%N E.g.f.: -LambertW(-x/(1 - x)) / (1 - x).
%F a(n) = Sum_{k=0..n-1} 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 = 18; CoefficientList[Series[-LambertW[-x/(1 - x)]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Binomial[n, k]^2 k! (n - k)^(n - k - 1), {k, 0, n - 1}], {n, 0, 18}]
%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, A052871, A245496, A277505, A331727.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 25 2020
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