Let \(a_k = (k^2 +1) k!\) and \(b_n = \displaystyle \sum_{k=1}^{n} a_k\).

\(\dfrac{a_{100}}{b_{100}}\) can be represented as \(\dfrac AB\) for positive coprime integers \(A\) and \(B\).

Find \(B-A\).

**Details and assumptions**

- \(n!\) is the factorial function

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