Generalised Pascal triangles and generalised sequences

[Ralf Stephan]

Further notes. My assessment of diagonal sums of (1) was wrong
and Paul's conjecture right---I confused C([n/j],k) with C([n/j],n-k).
The arguments however should be applicable if j is replaced with 2j,
as there is a change in projection.

> 2. The triangle binomial(floor((j*n-(j-1)*k-(j-1))/j,n-k) has row 
> sums with g.f. 1/(1-x-x^j) and diagonal sums with g.f.
> (1+x^(2j-1))/((1+x)(1-x-x^(2j)).

I think the g.f. is

B(x,y;j) = ((xy)^(j-1)-1)/(xy-1) + (xy)^(j-1)((xy)^j-1)/[(xy-1)(1-(xy)^j-x),

for similar reasons as with (1) but going into the xy direction.


Regards,
ralf