Find the sum of the series

Calculus Level 5

1+12+222!+12+22+323!+12+22+32+424!+\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 aeb\dfrac{ae}{b}, where aa and bb are coprime positive integers, find a+ba+b.

Clarification: e2.71828e \approx 2.71828 denotes the Euler's number.

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