[seqfan] Catalan triangles, polynomials and basket weaving

[David Scambler]

Allow me to draw to the attention of seqfans some interesting triangles.

The triangles have the following properties:

For all n>=1:
1) sum of coefficients = the nth Catalan number
2) sums of the anti-diagonals are the nth row of the Catalan/Narayana triangle A001263.
3) row sums are a row of an OEIS sequence.
4) column sums are a row of an OEIS sequence.

Here is an example for n=7 and the row sum sequence is A092107 and the column sum sequence is A091869.

1	12	38	44	25	6	1
9	48	67	32	5		
19	48	27	4			
16	16	3				
5	2					
1	

This implies that sequences A092107 and A091869 are related in a particular sense.

Here is an example where the row sum sequence is A091187 and the column sum sequence is A091867.	

0	1	3	21	10	15	0	1
0	12	36	52	20	6		
6	36	60	28	5			
12	40	24	4				
13	14	3					
4	2						
1							

I wonder whether anything can be said in general about pairs of OEIS (or other) sequences that
conform to this specification.

As it happens, T(i,j) in the examples above defines the coefficient of v^i*h^j in the polynomials
that count paths from (0,0) to (7,7) on basket weave (v,h)-tiling for the cases weakly and
strongly above the diagonal respectively.

In general, polynomials for paths from (0,0) to (n,k) on (v,h)-tiling weakly above y=x look like this:

n=1: 1, 
n=2: 1, 1 + h, 
n=3: 1, 1 + h + v, 1 + 2.h + h^2 + v, 
n=4: 1, 1 + 2.h + v, 1 + 3.h + h^2 + 2.v + v.h + v^2, 1 + 4.h + 3.h^2 + h^3 + 2.v + 2.v.h + v^2,

and for paths strongly above y=x

n=1: 1, 
n=2: 1, v, 
n=3: 1, v + h, v^2 + h, 
n=4: 1, 2.v + h, v + v^2 + h + h.v + h^2, v + v^3 + 2.h.v + h^2,


In an interesting development Paul Barry has found that these polynomial arrays can be derived using
 a technique similar to that used to process Riordan production matrices, in this case matrices of the form:

1, 1, 
h, h, 1, 
v, v, v, 1, 
h, h, h, h, 1, 
v, v, v, v, v, 1 
h, h, h, h, h, h, 1  

and

v, 1, 
h, h, 1, 
v, v, v, 1, 
h, h, h, h, 1, 
v, v, v, v, v, 1 
h, h, h, h, h, h, 1  

for the cases of paths weakly and strongly above y=x respectively.

Cheers
dave