I have two variable polynomials given by:
$$ a^{l,n}_{m}(X,Y) = \underset{k_1+k_2=l+n}{\sum_{k_1=-m}^{l+n} \sum_{k_2 = m}^{l -n +2m}} X^{k_1}Y^{k_2} = \sum_{k=0}^{l - n + m} X^{l + n - m - k} Y^{k + m}.$$
I was able to write this formula looking at the table representation:
I would also like to write the similar formula for next table:
In this case, I was able to figure out the case when $n=0$;
$$ a^{l,0}_{m}(X,Y) = \underset{k_1+k_2=l}{\sum_{k_1=0}^{l+m-1} \sum_{k_2 = 1}^{l+m+1}} X^{k_1}Y^{k_2}. $$ But I failed when trying to get the general case. How can I figure this out?
P.S. I can generate more terms in tables to check if results are correct.