I am doing some research in combinatorics, and I found that I have to consider the following binomial coefficient :

$$ \binom{\binom{i}{j}}{k} $$

(In fact, I have to take the product for fixed $i,k$ and odd $j$’s, but to make this product, I have to manipulate those coefficients, and I don’t know how.)

Is there a way to write it in terms of other binomial coefficients, power series, or other combinatorial tools?