Tricky Sum!

Algebra Level 4

n=21(n21)n2=134+189+11516+12425+ \sum_{n=2}^\infty \dfrac1{(n^2-1)n^2} = \dfrac{1}{3\cdot4}+\dfrac{1}{8\cdot 9}+\dfrac{1}{15\cdot16}+\dfrac{1}{24\cdot25}+\cdots

The series above is equal to ABπCD\dfrac{A-B\pi^{C}}{D} where A,B,CA,B,C and DD are positive integers with AA and BB coprime.

Find the value of A+B+C+DA+B+C+D.

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