Can you generalize this?

Algebra Level 5

113+2135+31357+ \dfrac1{1\cdot3} + \dfrac2{1\cdot3\cdot5} + \dfrac3{1\cdot3\cdot5\cdot7} + \cdots

If the sum above can be expressed as AB\dfrac AB, where BB is a prime and AA is a positive integer, find (A+B)!(A+B)!.

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