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

If the value of the sum above can be expressed in the form of $\dfrac{ae}{b}$, where $a$ and $b$ are coprime positive integers, find $a+b$.

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**Clarification:** $e \approx 2.71828$ denotes the Euler's number.