Another Combinatorial Summation!

\[\Large{H_{q,s} = \sum_{k=0}^s (-1)^{s+k} 2^{2k} \binom{q-1+k}{2k} \binom{q-1-k}{s-k} }\]

Generalize \(\large{H_{q,s}}\) in terms of \(q\) and \(s\), and then evaluate the value of \(\large{H_{15,10}}\).

Note: You may use a Calculator for the final calculation. Don't use the Scratch Pad.

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