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URL: https://oeis.org/A221864

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A221864
Number of forests (sets) of rooted labeled trees on {1,2,...,n} such that the node with label 1 is in the same rooted tree as the node with label 2.
2
0, 0, 2, 11, 88, 930, 12254, 193736, 3576564, 75552560, 1797906490, 47601571968, 1388102588048, 44210926113536, 1527152437488150, 56867807937459200, 2271048787266451756, 96826981390532388864, 4389830567318703987314, 210886652765343862784000
OFFSET
0,3
LINKS
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
Sequence in context: A361599 A138739 A216831 * A376871 A197900 A106961
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Apr 10 2013
STATUS
approved