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E.g.f.: -x - LambertW(-x/(1+x)) - (x/2)*LambertW(-x/(1+x))^2.
(PARI) seq(n)={my(w=lambertw(-x/(1+x) + O(x*x^n))); Vec(serlaplace(-x - w - (x/2)*w^2), -n)} \\ Andrew Howroyd, Dec 09 2020
From Andrew Howroyd, Dec 09 2020: (Start)
a(n) = Sum_{k=1..n} (-1)^(n-k)*k^(k-2)*n*(n-2)!*binomial(n-1,k-1)*(2*k*n - n - k^2)/k!) for n > 1.
(End)
(PARI) a(n) = {if(n<=1, 0, sum(k=1, n, (-1)^(n-k)*k^(k-2)*n*(n-2)!*binomial(n-1, k-1)*(2*k*n - n - k^2)/k!))} \\ Andrew Howroyd, Dec 09 2020
0, 0, 0, 4, 5, 186, 847, 17928, 166833, 3196630, 45667391, 925287276, 17407857337, 393376875906, 8989368580935, 229332484742416, 6094576250570849, 174924522900914094, 5271210321949744111, 168792243040279327860, 5674164658298121248361, 200870558472768096534490
Andrew Howroyd, <a href="/A331578/b331578.txt">Table of n, a(n) for n = 1..200</a>
nonn,more
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Terms a(9) and beyond from Andrew Howroyd, Dec 09 2020
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Number of labeled series-reduced rooted trees with n vertices and more than two branches of the root.
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