1,2

a(n)= sum N/D over all the partitions of n:1K1+2K2+ ... + nKn, with smallest part greater than 1, where N = n!*product_{1=<i<=n}i^((i-1)Ki) and D = product_{1=<i<=n}(Ki!(i!)^Ki).

E.g.f.: -exp(-x)*LambertW(-x)/x. a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(k+1)^(k-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 22 2005

a(0) = 1, a(n) = Sum_{j=1..n-1} C(n-1,j) (j+1)^j a(n-1-j) if n>0. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]

a(5)= 805 because there are 625 such trees and 5 vertices can be partitioned in two trees only in one way: 3 go to one tree and 2 go to the other. It's impossible to split 5 vertices in 3 or more trees without give only one vertex to a tree. Each one of the 3^2 distinct trees on 3 vertices can be labeled in C(5, 3) manners and to each one of the 9C(5, 3) = 90 possibilities there are 2 different trees of order 2, so we get 180 forests of two trees.

a:= proc(n) option remember; if n=0 then 1 else add (binomial (n-1, j) *(j+1)^j *a(n-1-j), j=1..n-1) fi end: seq (a(n), n=1..25); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]

Cf. A033185, A105599.

Sequence in context: A015473 A029849 A080638 this_sequence A123680 A132621 A108992

Adjacent sequences: A105782 A105783 A105784 this_sequence A105786 A105787 A105788

nonn

Washington Bomfim (webonfim(AT)bol.com.br), Apr 21 2005

More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008