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.

0, 0, 2, 11, 88, 930, 12254, 193736, 3576564, 75552560, 1797906490, 47601571968, 1388102588048, 44210926113536, 1527152437488150, 56867807937459200, 2271048787266451756, 96826981390532388864, 4389830567318703987314, 210886652765343862784000

Offset: 0

Author

Geoffrey Critzer, Apr 10 2013

Keywords

• nonn

Examples

			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 '.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000169, A000272.

Programs

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 *)

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