Perfect triangular numbers

If we pick any even perfect number less than $9^{9^{9^{9^{9^9}}}}$ at random, what is the probability that the perfect number chosen is also a triangular number?

Express the probability as a rational number $\dfrac ab$ , where $a$ and $b$ are coprime positive integers . And submit $\mu\left(a+b\right)\cdot\left(ab\right)^2$ as your answer.

Notation : $\mu$ denotes the Möbius function .