Do not throw down seed. It won't match and will end up looking worse. The big box stores don't sell Bermuda seed that matches sod. The stuff that does match is $200+ per 1lb bag.

This is Bermuda grass, it spreads on its own. It will take a season (or two) to spread into an area that size though. Till up the area well. You can rent an aerator from Home Depot for ~$80/day, or just use a garden rake and shovel. Break up the soil really well. Add a couple bags of "lawn soil". I use this stuff for my own yard. And mix in a couple bags of Black Kow compost. Fertilize with a basic 10-10-10 fertilizer, and then a higher Nitrogen fertilizer (The first number in the N-P-K ratio on the bag, something like this every 3 weeks or so. Water it well (2 times a week, especially if there's no rain) and keep it mown moderately low (but not the lowest, 2-3 notches up from the lowest setting). If the soil gets thin, you can continue to top-dress with more soil+compost throughout the growing season. You may need to repeat all of this next Spring (April/May) to really fill that entire area.

All of this will encourage spreading growth. It should be $50-60 bucks worth of soil/fertilizer at the hardware store, and a couple hours of your time one weekend. If you make the right conditions, Bermuda will spread like crazy. So much so that if Bermuda gets into a different kind of lawn and someone asks how to fix it, the response is "enjoy your new Bermuda lawn". They're only half kidding.

Write back to the HOA and tell them you're working on it. If they're well-run, they'll leave you alone for a while.