Generating symmetric pascal matrix

Question

Symmetric pascal matrix defined as $S=\binom{i+j}{i}$ , show that if $i=j$ by matrix multiplication we have $$S^3=\sum_{k_1 , k_2 =0}^{q^\alpha-1} \binom{i+k_1}{i}\binom{k_1 +k_2}{k_1}\binom{k_2 + j}{k_2}=q^2\varepsilon \binom{i+j}{i} +1$$ where $q$ is prime, $\alpha ,\varepsilon \in \mathbb{Z}^+$.