Sum it again

Calculus Level 3

1+222!+323!+424!+\large 1+\frac{2^{2}}{2!}+\frac{3^{2}}{3!}+\frac{4^{2}}{4!}+\ldots

If the series above equals to n×e n\times e for e=limx0(1+1x)x\displaystyle e =\lim_{x\to0} \left(1+\frac1x\right)^x , find the value of nn.

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