Extend New Seqs: "k-Convoluted Trees" Counts

[Martin Fuller]

Paul, seqfans,

I have attached a PARI program (with explanation) which I used to
generate the values below. They match your calculations except for the
6th and 7th terms of A132855.

A132851: 
Number of nodes at generation n in the 2-convoluted tree (A132850). 
1, 1, 2, 4, 14, 62, 462, 5380, 105626, 3440686, 196429906, 19603795552,
3496015313038, 1120368106124268, 653253602487886098,
697073727912597623594, 1371575342274982257650434,
5001872822460132255638199998, 33985054727503111175373886399250,
432024026653870819584750328953621778

A132853: 
Number of nodes at generation n in the 3-convoluted tree (A132852). 
1, 1, 3, 18, 180, 4347, 245511, 33731424, 11850958449, 10823718435525,
26127739209077469, 169071160476526474689, 2962647736390311022542681,
141814999458311839862777779311, 18682218330844513414826192858258922,
6816346360277755893118363665630012225420

A132855: 
Number of nodes at generation n in the 4-convoluted tree (A132854). 
1, 1, 4, 32, 736, 47600, 9901728, 6780161344, 15819971230848,
128391245362464512, 3685238521747987153664,
378871127417706380405937152, 140962622184196263047081802452992,
191428155805533938524028481989647915008,
955702499453836874538617308649867009480896512

A132857: 
Number of nodes at generation n in the 5-convoluted tree (A132856). 
1, 1, 5, 75, 3625, 638750, 442823125, 1278820631250, 15775429658296875,
848938273203627578125, 202483260558673741179296875,
216741216953142470752123517187500,
1051774892873652266440974611041742187500,
23332485704169236846450189449001711184697265625

A132859: 
Number of nodes at generation n in the 6-convoluted tree (A132858). 
1, 1, 6, 108, 7614, 2451762, 3773520918, 28927494486144,
1137959521626242430, 234471053096681379609150,
257075108927481255273258364890, 1518584605077301579030226106654776268,
48819910122176867311132781943952677374210562,
8612868429525484388625401466547224861259048650708888

Martin Fuller

--- "Paul D. Hanna" <pauldhanna at juno.com> wrote:

> 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. 
> >  
> > ...
> >   
> > 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. 
> >  
> > ------------------------------------------------------
> [...]
> 
-------------- next part --------------
An embedded and charset-unspecified text was scrubbed...
Name: A132860.gp.txt
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20070925/d4a9934c/attachment-0001.txt>