What is the ruling on the prayer of an Imam who forgot the opening Takbir (Takbirat al-Ihram) then performed the two prostrations of forgetfulness (Sajdat as-Sahw), and what is the ruling on the prayer of those praying behind him?

The opening takbir (Allahu Akbar) is a pillar of prayer, and the prayer is not valid without it. Therefore, whoever omits it intentionally or forgetfully, their prayer is invalid and they must repeat it. Similarly, the congregants who followed him must repeat their prayer, because they followed someone who did not even enter the prayer in the first place.