W. Lang, Aug 07 2007

 A131449 
 
 Number of organic (also called increasing) labellings of rooted ordered trees.
 
 Organic labelling means that the vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing.

 The length of row n is C(n):=A000108(n) (Catalan numbers): [1,1,2,5,14,42,...].

 Isomorphic rooted trees are separated by vertical bars. The number of rooted trees with n non-root vertices is [1, 1, 2, 4, 9, 20,...]=A000081(n+1), n>=0.

 The row sums give [1,1,3,15,105,945,..] = A001147(n).

 A035342(n,1), first column of S2(3) triangle.


 For the rooted ordered trees with n=0,..,5 non-root vertices see the W. Lang link.
     
  
 n\m    1   2  3   4   5   6   7  8     9 10  11   12  13   14   15 16 17  18 19   20 21 22 23   24 25 26   27 28   29 30  31 32  33  34  35 36  37  38  39 40  41  42 
  
  0     1   

  1     1   

  2     2 | 1
        
  3     6 | 3  3 | 2 | 1

  4    24 |12 12  12 | 8   8 | 6 | 6 |  4  4 | 3    3 | 2 |  1
     
  5   120 |60 60  60  60 |40  40  40 | 30 30  30 | 30  30 | 24 | 20 20 20 |20 20 | 15 15 15 15 | 12 12 12 | 10 10 | 10 10 | 8  8 | 6 | 6 | 5  5 | 4   4 | 3  3 | 2 | 1   

 
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  n\m    1   2  3   4   5   6   7  8     9 10  11   12  13   14   15 16 17  18 19   20 21 22 23   24 25 26   27 28   29 30  31 32  33  34  35 36  37  38  39 40  41  42



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