It is critical to seed as early as possible within this window. Even when seeding within this window, waiting one week later to seed may mean the turf will take 2 more weeks to mature as germination and subsequent growth slows later in the fall as temperatures cool. Additionally, annual bluegrass and other winter annual weeds will be more problematic with fall seeding dates than late summer seeding dates.
Seeding early (later summer) allows the turf to maximize its establishment and rooting prior to the next summer's heat and drought.
For more information about establishment, see:
Aaron Patton, Turfgrass Extension Specialist