.

Bulb - socket - ballast cord - ballast - generator
This is how they arrange and connect to each other. You will have to wire the [socket] to [ballast cord] (which is very easy and fairly basic science) unless you can find one that is already connected.
You said that the ballast cord will fit the ballast why do you need the cord? The ballast also doesn't fit the electrical outlet on a generator. 

 
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