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$\large 1+\frac{2^{2}}{2!}+\frac{3^{2}}{3!}+\frac{4^{2}}{4!}+\ldots$

If the series above equals to $n\times e$ for $\displaystyle e =\lim_{x\to0} \left(1+\frac1x\right)^x$, find the value of $n$.

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