I don't have time to write a detailed answer but the following should help.
Blast your sum with Newton's binomial formula. That gives a sum over three indices $i,j,k$. The three equations you get on these indices by fixing the monomial of interest are linearly dependent, so you are left with a sum over just one index say $i$. I believe what you get is ${}_3F_{2}$ hypergeometric sum. Your quantity of interest is also a Wigner $3jm$-symbol or Clebsch-Gordan coefficient in the literature on quantum angular momentum. In general determining the zeros of such numbers is an open problem, see my article "The bipartite Brill-Gordan locus and angular momentum" with Jaydeep Chiplakatti for some references. If your case is sufficiently special (e.g., stretched coefficients) then you might be lucky enough to know if your coefficients are zero or not.