Extend New Seqs: "k-Convoluted Trees" Counts

[Paul D. Hanna]

Seqfans, 
    Would someone extend the following NEW sequences, please. 
These sequences count the number of nodes in generation n of 
"k-convoluted trees" at k=2,3,4,5, and 6. 
Below I define "k-convoluted trees", and provide examples. 
 
I have started the sequences, but would like more terms. 
These also need verifying, since done by hand! 
 
The 6 sequences needing extending begin as follows. 
 
A132851: 
Number of nodes at generation n in the 2-convoluted tree (A132850). 
1, 1, 2, 4, 14, 62, 462, 
  
A132853: 
Number of nodes at generation n in the 3-convoluted tree (A132852). 
1, 1, 3, 18, 180, 4347, 
 
A132855: 
Number of nodes at generation n in the 4-convoluted tree (A132854). 
1, 1, 4, 32, 736, 
 
A132857: 
Number of nodes at generation n in the 5-convoluted tree (A132856). 
1, 1, 5, 75, 3625, 
 
A132859: 
Number of nodes at generation n in the 6-convoluted tree (A132858). 
1, 1, 6, 108, 7614, 
 
A132860: 
Table, read by antidiagonals, where row k gives the number of nodes 
in generation n, n>=0, of the k-convoluted tree for k>=1. 
1, 1, 1, 1, 1, 1, 1, 1, ... 
1, 1, 2, 4, 14, 62, 462, ...
1, 1, 3, 18, 180, 4347, ...
1, 1, 4, 32, 736, ...
1, 1, 5, 75, 3625, ...
1, 1, 6, 108, 7614, ...
...
  
Thanks, 
    Paul 
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DEFINITION: k-Convoluted Tree. 
 
Tree of all finite sequences {a(i), i=0..n} that form the initial terms 
of a self-convolution k-th power of some integer sequence such that 
0 < a(n) <= k*a(n-1) for n>0 with a(0)=1. 
 
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CASE k=2: the 2-convoluted tree.

ID#: A132850
Tree of all finite sequences {a(k), k=0..n} that form the initial terms
of 
a self-convolution square of some integer sequence such that 
0 < a(n) <= 2*a(n-1) for n>0 with a(0)=1. 

THE 2-CONVOLUTED TREE. 
Generations 0..5 of the 2-convoluted tree are as follows;  
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1] ;
GEN.1: 1->[2] ;
GEN.2: 1-2->[1,3] ;
GEN.3: 
1-2-1->[2]
1-2-3->[2,4,6] ;
GEN.4:
1-2-1-2->[2,4]
1-2-3-2->[1,3]
1-2-3-4->[1,3,5,7]
1-2-3-6->[1,3,5,7,9,11] ;
GEN.5:
1-2-1-2-2->[2,4]
1-2-1-2-4->[2,4,6,8]
1-2-3-2-1->[2]
1-2-3-2-3->[2,4,6]
1-2-3-4-1->[2]
1-2-3-4-3->[2,4,6]
1-2-3-4-5->[2,4,6,8,10]
1-2-3-4-7->[2,4,6,8,10,12,14]
1-2-3-6-1->[2]
1-2-3-6-3->[2,4,6]
1-2-3-6-5->[2,4,6,8,10]
1-2-3-6-7->[2,4,6,8,10,12,14]
1-2-3-6-9->[2,4,6,8,10,12,14,16,18]
1-2-3-6-11->[2,4,6,8,10,12,14,16,18,20,22].
Each path in the tree from the root node forms the initial terms 
of a self-convolution square sequence of integer terms.  

The minimal path in the 2-convoluted tree is A083952 and 
the maximal path is A132831. 
------------------------------------------------------

CASE k=3: the 3-convoluted tree.

ID#: A132852
Tree of all finite sequences {a(k), k=0..n} that form the initial terms 
of a self-convolution cube of some integer sequence such that 
0 < a(n) <= 3*a(n-1) for n>0 with a(0)=1. 

THE 3-CONVOLUTED TREE. 
Generations 0..4 of the 3-convoluted tree are as follows;  
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1] ;
GEN.1: 1->[3] ;
GEN.2: 1-3->[3,6,9] ;
GEN.3: 
1-3-3->[1,4,7]
1-3-6->[1,4,7,10,13,16] 
1-3-9->[1,4,7,10,13,16,19,22,25] ;
GEN.4:
1-3-3-1->[3]
1-3-3-4->[3,6,9,12]
1-3-3-7->[3,6,9,12,15,18,21]
1-3-6-1->[3]
1-3-6-4->[3,6,9,12] 
1-3-6-7->[3,6,9,12,15,18,21]
1-3-6-10->[3,6,9,12,15,18,21,24,27,30]
1-3-6-13->[3,6,9,12,15,18,21,24,27,30,33,36,39]
1-3-6-16->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48]
1-3-9-1->[3]
1-3-9-4->[3,6,9,12] 
1-3-9-7->[3,6,9,12,15,18,21]
1-3-9-10->[3,6,9,12,15,18,21,24,27,30]
1-3-9-13->[3,6,9,12,15,18,21,24,27,30,33,36,39]
1-3-9-16->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48]
1-3-9-19->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57]
1-3-9-22->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66
]
1-3-9-25->[3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66
,69,72,75].
Each path in the tree from the root node forms the initial terms of 
a self-convolution cube sequence of integer terms.  

The minimal path in the 3-convoluted tree is A083953 and 
the maximal path is A132835.
------------------------------------------------------

CASE k=4: the 4-convoluted tree.

ID#: A132854
Tree of all finite sequences {a(k), k=0..n} that form the initial terms 
of a self-convolution 4-th power of some integer sequence such that 
0 < a(n) <= 4*a(n-1) for n>0 with a(0)=1. 

THE 4-CONVOLUTED TREE. 
Generations 0..3 of the 4-convoluted tree are as follows;  
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1] ;
GEN.1: 1->[4] ;
GEN.2: 1-4->[2,6,10,14] ;
GEN.3: 
1-4-2->[4,8]
1-4-6->[4,8,12,16,20,24] 
1-4-10->[4,8,12,16,20,24,28,32,36,40] 
1-4-14->[4,8,12,16,20,24,28,32,36,40,44,48,52,56].
Each path in the tree from the root node forms the initial terms of 
a self-convolution 4-th power sequence of integer terms.  
 
The minimal path in the 4-convoluted tree is A083954 and 
the maximal path is A132837.  
------------------------------------------------------

CASE k=5: the 5-convoluted tree.

ID#: A132856
Tree of all finite sequences {a(k), k=0..n} that form the initial terms 
of a self-convolution 5-th power of some integer sequence such that 
0 < a(n) <= 5*a(n-1) for n>0 with a(0)=1. 

THE 5-CONVOLUTED TREE. 
Generations 0..3 of the 5-convoluted tree are as follows;  
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1] ;
GEN.1: 1->[5] ;
GEN.2: 1-5->[5,10,15,20,25] ;
GEN.3: 
1-5-5->[5,10,15,20,25]
1-5-10->[5,10,15,20,25,30,35,40,45,50] 
1-5-15->[5,10,15,20,25,30,35,40,45,50,55,60,65,70,75] 
1-5-20->[5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100]
1-5-25->[5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,105,
110,115,120,125].
Each path in the tree from the root node forms the initial terms of 
a self-convolution 5-th power sequence of integer terms.  
 
The minimal path in the 5-convoluted tree is A083955 and 
the maximal path is A132839.  
------------------------------------------------------

CASE k=6: the 6-convoluted tree.

ID#: A132858
Tree of all finite sequences {a(k), k=0..n} that form the initial terms 
of a self-convolution 6-th power of some integer sequence such that 
0 < a(n) <= 6*a(n-1) for n>0 with a(0)=1. 

THE 6-CONVOLUTED TREE. 
Generations 0..3 of the 6-convoluted tree are as follows;  
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1] ;
GEN.1: 1->[6] ;
GEN.2: 1-6->[3,9,15,21,27,33] ;
GEN.3: 
1-6-3->[2,8,14]
1-6-9->[2,8,14,20,26,32,38,44,50] 
1-6-15->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86] 
1-6-21->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122
]
1-6-27->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122
,128,134,140,146,152,158]
1-6-33->[2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122
,128,134,140,146,152,158,164,170,176,182,188,194].
Each path in the tree from the root node forms the initial terms of 
a self-convolution 6-th power sequence of integer terms.  
 
The minimal path in the 6-convoluted tree is A083956.
   
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END.