Three Square Series

. $\frac {4}{1} + \frac {25}{25} + \frac {64}{144} + \frac {121}{484} + \frac {196}{1225} \cdots = \left(\frac {A}{2} \cdot \frac {4\cdot36\cdot100\cdot196\cdot324\cdots}{3\cdot35\cdot99\cdot195\cdot323\cdots} \cdot \frac {9\cdot81\cdot225\cdot441\cdot729\cdots}{8\cdot80\cdot224\cdot440\cdot728\cdots} \right)^2$

Report $A$ to 5 decimal places.