Lozanic's triangle A034851 -- Matrix Log Holds Key

[Paul D. Hanna]

Christian (and Seqfans), 
        Using the trend observed in the matrix log, I calculated more
rows of your triangle. 
If your original triangle continues as:  
1;
1,1;
1,0,1;
1,1,1,1;
1,0,2,0,1;
1,1,2,2,1,1;
1,0,3,0,3,0,1;
1,1,3,3,3,3,1,1;
1,0,4,0,6,0,4,0,1;
1,1,4,4,6,6,4,4,1,1;
1,0,5,0,10,0,10,0,5,0,1;
1,1,5,5,10,10,10,10,5,5,1,1; ...
 
then this triangle is already in the OEIS as A051159 ...
is that correct?    
 
At any rate, the inverse is not in the OEIS, and is worthy.  
 
Also, it may be interesting to note the following related sequences. 
 
Antidiagonal sums = A053602  
a(n)=a(n-1)-(-1)^n*a(n-2), a(0)=0, a(1)=1.  
1,1,2,1,3,2,5,3,8,5,13,8,21,13,34,21,55,34,89,55,144,
 
Row squared sums = A063886  
Number of n-step walks on a line starting from the origin but not
returning to it.  
1,2,2,4,6,12,20,40,70,140,252,504,924,1848,3432,6864,
 
Paul
 
On Sat, 4 Mar 2006 00:29:57 -0500 "Paul D. Hanna" <pauldhanna at juno.com>
writes:
> Dear Christian (and Seqfans)
>          Very interesting triangle ... when one takes the matrix log, 
> the pattern is obvious and significant.  
> This pattern accounts for the unusual sign pattern in the matrix 
> inverse:
> log(T) = 
> 0;
> 1, 0;
> 1, 0, 0;
> 0, 1, 1, 0;
> 0, 0, 2, 0, 0;
> 0, 0, 0, 2, 1, 0;
> 0, 0, 0, 0, 3, 0, 0;
> 0, 0, 0, 0, 0, 3, 1, 0; ...
>  
> This pattern also says something about the e.g.f of your original
> triangle. 
>   
> Paul
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