I have a 13,500 air conditioner and a 4000 watt generator. Will this gennie pull this a/c?

If it is a built-in Onan or Kohler, yes.

If it is a Japanese portable, yes.

If it is a Chinese portable, maybe.

Stupid reminder time: Always make sure the AC is off when starting the genny, and the genny is fully powered up before starting the AC. There should be no noticable drag [hesitation], from the AC compressor. If it seems to be struggling to come up to speed, you could very well be damaging the unit. If it cranks right up and blows cool, no problem. The genny will, of course, sound like it's working harder, it is! Others will clarify and correct me here, and I will be grateful for it.

4000 watts is 30 to 35 amp rating for 115VAC so there should be no problem. There may be little left over to pop popcorn in the microwave while you watch the tube and the frig comes on because you had the door open too long.