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A048816 Number of rooted trees with n nodes with every leaf at the same height.

%I A048816 #39 May 15 2019 05:54:58
%S A048816 1,1,2,3,5,7,12,17,28,42,68,103,168,260,420,665,1075,1716,2787,4489,
%T A048816 7304,11849,19333,31504,51561,84347,138378,227096,373445,614441,
%U A048816 1012583,1669774,2756951,4555183,7533988,12469301,20655523,34238310,56795325,94270949
%N A048816 Number of rooted trees with n nodes with every leaf at the same height.
%C A048816 The trees are unordered (see A000081). For balanced ordered rooted trees see A079500. - _Joerg Arndt_, Jul 20 2014
%C A048816 The trees are unlabeled. For labeled version see A238372. - _Alois P. Heinz_, Dec 29 2014
%H A048816 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 A048816 Joerg Arndt, <a href="/A048816/a048816.txt">balanced unordered rooted trees for n = 1..10</a>
%H A048816 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%e A048816 See Arndt link.
%e A048816 From _Gus Wiseman_, Oct 08 2018: (Start)
%e A048816 The a(1) = 1 through a(7) = 12 balanced rooted trees with n nodes:
%e A048816   o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
%e A048816           ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
%e A048816                  (((o)))  (((oo)))   (((ooo)))    (((oooo)))
%e A048816                           ((o)(o))   ((o)(oo))    ((o)(ooo))
%e A048816                           ((((o))))  ((((oo))))   ((oo)(oo))
%e A048816                                      (((o)(o)))   ((((ooo))))
%e A048816                                      (((((o)))))  (((o)(oo)))
%e A048816                                                   ((o)(o)(o))
%e A048816                                                   (((((oo)))))
%e A048816                                                   ((((o)(o))))
%e A048816                                                   (((o))((o)))
%e A048816                                                   ((((((o))))))
%e A048816 (End)
%t A048816 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 A048816 Cf. A048808-A048815.
%Y A048816 Row sums of A244925.
%Y A048816 Cf. A079500, A238372.
%Y A048816 Cf. A000081, A000311, A001678, A120803, A320154, A320160, A316624, A320169.
%K A048816 nonn
%O A048816 1,3
%A A048816 _Christian G. Bower_, Apr 15 1999