Combinatorial Number Theory!

\[\large{4^s \sum_{k=0}^{n-s} { 2s+2k-1 \choose 2s-1} \equiv \alpha \pmod p}\]

Let \(p=2n+1\) be a prime, where \(n\) is an integer, and let \(s\) be any integer such that \(1 \leq s \leq n\). If the above modular equation satisfies, where \(\alpha\) is a positive integer less than \(p\), find the value of \(\alpha\).

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