OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..n-2} C(n-2,k)*(k+2)^(k+1)*(n-k-1)^(n-k-3).
E.g.f.: Double integral of T''(x)*exp(T(x)) dx^2 where T(x) is the e.g.f. for A000169.
a(n) ~ exp(1) * n^(n-1) * (1 - sqrt(Pi/(2*n))). - Vaclav Kotesovec, Aug 31 2014
EXAMPLE
a(3) = 9 + 2 = 11 because we have A000169(3) = 9 forests composed of a single rooted tree and 2 forests composed of two rooted trees:
1'-2 3, 2'-1 3 where the root is indicated with '.
MAPLE
a:= n-> add(binomial(n-2, k)*(k+2)^(k+1)*(n-k-1)^(n-k-3), k=0..n-2):
seq(a(n), n=0..20); # Alois P. Heinz, Apr 11 2013
MATHEMATICA
nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ Series[Integrate[Integrate[D[D[t, x], x]Exp[t], x], x], {x, 0, nn}], x]
Flatten[{0, 0, CoefficientList[Series[-(2 + LambertW[-x]) / (x^3*(1 + 1/LambertW[-x])^3), {x, 0, 20}], x] * Range[0, 20]!}] (* Vaclav Kotesovec, Aug 31 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Apr 10 2013
STATUS
approved