> 1. The triangle binomial(floor(n/j), n-k) has row sums with g.f.
> sum{i=0..j-1, x^i}/(1-2x^j), and diagonal sums with g.f.
> sum{i=0..j-1, x^(2i)}/(1-x^(2j-1)-x^(2j)).
If it helps, the o.g.f. of that is (unproved) x^j/[(x-1)(1-x^j(1+y))].
ralf