Wolstenholme's Theorem, Vandermonde's Convolution Identity and so much more

Let \(p=2^{16}+1\) be an odd prime. Define \(\large{H_{n}=\sum _{ x=1 }^{ n }{ \frac { 1 }{ n } } }\). Find the remainder when

\[\Large{\left( p-1 \right) !\sum _{ n=1 }^{ p-1 }{ { H }_{ n } } \cdot { 4 }^{ n }\cdot \left( \begin{matrix} 2p-2n \\ p-n \end{matrix} \right) }\]

is divided by \(p\)

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