Extend New Seqs: "k-Convoluted Trees" Counts

[Paul D. Hanna]

Seqfans, 
     OK, using PARI program code I have confirmed the counts 
and extended the sequences a few terms.  
Here are my results so far (immediately below). 
 
I now have enough terms to submit to OEIS, 
but can anyone extend them further ? 
in particular, the 2- and 3-convoluted tree counts. 
  
Thanks, 
      Paul 
  
A132851: 
Number of nodes at generation n in the 2-convoluted tree (A132850). 
1, 1, 2, 4, 14, 62, 462, 5380, 105626, 

A132853: 
Number of nodes at generation n in the 3-convoluted tree (A132852). 
1, 1, 3, 18, 180, 4347, 245511,

A132855: 
Number of nodes at generation n in the 4-convoluted tree (A132854). 
1, 1, 4, 32, 736, 23936, 1261024, 

A132857: 
Number of nodes at generation n in the 5-convoluted tree (A132856). 
1, 1, 5, 75, 3625, 638750, 

A132859: 
Number of nodes at generation n in the 6-convoluted tree (A132858). 
1, 1, 6, 108, 7614, 2451762, 
 
END. 
 
On Tue, 18 Sep 2007 01:49:14 -0400 "Paul D. Hanna" <pauldhanna at juno.com>
writes:
> 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 
> ------------------------------------------------------
>  
> 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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