Pi sum

Calculus Level 3

\[\large \lim _{ n\rightarrow \infty }{ \frac { 7\sqrt { n } }{ { 4 }^{ n } } \sum _{ k=0 }^{ n }{ \begin{pmatrix} n \\ k \end{pmatrix}^{ 2 } } } =\frac { a }{ \sqrt { \pi } } \]

Find the integer \(a\) that satisfies the equation above. If you think that such integer does not exist, enter your answer as 666.

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