A225573 Number of trees over all forests of labeled rooted trees in which some (possibly all or none) of the trees have been specially designated.

0, 2, 12, 90, 864, 10290, 147456, 2480058, 48000000, 1052307234, 25798901760, 699896958618, 20826335158272, 674680957031250, 23643898043695104, 891412022961534330, 35982083287879778304, 1548474957047229426498, 70778880000000000000000

Offset: 0

Author

Geoffrey Critzer, May 10 2013

Keywords

• nonn

Comments

The expected number of trees in each such forest a(n)/A089140(offset) approaches 3 as n gets large.

Crossrefs

Cf. A225497.

Programs

Mathematica

nn = 18; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Range[0, nn]! CoefficientList[Series[D[Exp[y tx]^2, y] /. y -> 1, {x, 0, nn}], x]

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*n^(n-k)*2^k*k = 6*n*(n+2)^(n-2).