%I #39 May 15 2019 05:54:58

%S 1,1,2,3,5,7,12,17,28,42,68,103,168,260,420,665,1075,1716,2787,4489,

%T 7304,11849,19333,31504,51561,84347,138378,227096,373445,614441,

%U 1012583,1669774,2756951,4555183,7533988,12469301,20655523,34238310,56795325,94270949

%N Number of rooted trees with n nodes with every leaf at the same height.

%C The trees are unordered (see A000081). For balanced ordered rooted trees see A079500. - _Joerg Arndt_, Jul 20 2014

%C The trees are unlabeled. For labeled version see A238372. - _Alois P. Heinz_, Dec 29 2014

%H Vaclav Kotesovec, <a href="/A048816/b048816.txt">Table of n, a(n) for n = 1..3500</a> (terms 1..300 from Alois P. Heinz)

%H Joerg Arndt, <a href="/A048816/a048816.txt">balanced unordered rooted trees for n = 1..10</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%e See Arndt link.

%e From _Gus Wiseman_, Oct 08 2018: (Start)

%e The a(1) = 1 through a(7) = 12 balanced rooted trees with n nodes:

%e o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)

%e ((o)) ((oo)) ((ooo)) ((oooo)) ((ooooo))

%e (((o))) (((oo))) (((ooo))) (((oooo)))

%e ((o)(o)) ((o)(oo)) ((o)(ooo))

%e ((((o)))) ((((oo)))) ((oo)(oo))

%e (((o)(o))) ((((ooo))))

%e (((((o))))) (((o)(oo)))

%e ((o)(o)(o))

%e (((((oo)))))

%e ((((o)(o))))

%e (((o))((o)))

%e ((((((o))))))

%e (End)

%t T[n_, k_] := T[n, k] = If[n==1, 1, If[k==0, 0, Sum[Sum[If[d<k, 0, T[d, k-1] * d], {d, Divisors[j]}]*T[n-j, k], {j, 1, n-1}]/(n-1)]]; a[n_] := Sum[ T[n, k], {k, 0, n-1}]; Array[a, 40] (* _Jean-François Alcover_, Jan 08 2016, after _Alois P. Heinz_ *)

%Y Cf. A048808-A048815.

%Y Row sums of A244925.

%Y Cf. A079500, A238372.

%Y Cf. A000081, A000311, A001678, A120803, A320154, A320160, A316624, A320169.

%K nonn

%O 1,3

%A _Christian G. Bower_, Apr 15 1999