Is there a triangle array that takes the preceding number and adds one for the first and subtracts one for the second term?

Question

I was wondering if there is a triangle array that would produce

0
1,-1
2,0,0,-2
3,1,1,-1,1,-1,-1,-3
4,2,2,0,2,0,0,-2,2,0,0,-2,0,-2,-2,-4

where i becomes i+1, i-1

I'm looking for a way to determine if @ row x and position y the value is 0.