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Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly increasing.
2

%I #10 Aug 31 2014 02:00:41

%S 1,1,2,3,6,9,20,31,67,115,238,406,880,1494,3118,5568,11408,20092,

%T 41938,73687,151529,272359,552611,987277,2033173,3617167,7371745,

%U 13318869,26991289,48496985,99264686,177917588,362349258,655925568,1331038563,2401097768,4913906801,8844673793,18046697901,32720071992,66666578597

%N Number of rooted ordered trees with n non-root nodes such that the branch lengths are weakly increasing.

%H Joerg Arndt, <a href="/A246565/a246565.txt">All trees for 0 <= n <= 7</a>

%e The a(4) = 6 such trees are:

%e :

%e : 1:

%e : [ 1 1 1 1 ]

%e : [ 0 0 0 0 ]

%e :

%e : O--o

%e : .--o

%e : .--o

%e : .--o

%e :

%e :

%e : 2:

%e : [ 1 1 2 ]

%e : [ 0 0 0 ]

%e :

%e : O--o

%e : .--o

%e : .--o--o

%e :

%e :

%e : 3:

%e : [ 1 3 ]

%e : [ 0 0 ]

%e :

%e : O--o

%e : .--o--o--o

%e :

%e :

%e : 4:

%e : [ 2 2 ]

%e : [ 0 0 ]

%e :

%e : O--o--o

%e : .--o--o

%e :

%e :

%e : 5:

%e : [ 2 2 ]

%e : [ 0 1 ]

%e :

%e : O--o--o

%e : .--o--o

%e :

%e :

%e : 6:

%e : [ 4 ]

%e : [ 0 ]

%e :

%e : O--o--o--o--o

%e :

%e See the Arndt link for all trees for 0 <= n <= 7.

%Y Cf. A000108 (all trees), A246566 (descending branch lengths), A193196, A192243.

%K nonn

%O 0,3

%A _Joerg Arndt_, Aug 30 2014