The only rule that the AKC has regarding the registering of litters is that both of the parents of the puppies must be registered dogs of the same breed. The relationship of those dogs to each other has no bearing at all on whether or not the litter is registerable.
What you need to keep in mind, however, is that breeding a dog to one of its own offspring is a VERY close inbreeding. Inbreeding is something that only breeders who know every dog in their bloodlines inside and out should attempt, because while it can be an invaluable tool in cementing desirable characteristics in a bloodline, if there are any hidden genetic problems in any of the dogs in that line, inbreeding is going to produce a litter of ticking timebombs just waiting to go off. Please don't take offense at what I am going to say, but there is really no other way to say this - if you had to ask the question about the AKC allowing the registration of puppies resulting from a parent/puppy breeding, you do not have the knowledge of breeding necessary to even consider inbreeding. The results will be disastrous. If you are inquiring because you are considering purchasing a puppy that is the result of one of these breedings and want to know if the registration papers will be legitimate, yes, they will, but I would think twice and three times and probably not purchase a puppy from one of these breedings, since you could end up with a genetic ticking time bomb.
Realize, also, that total outcrossing can have results that are bad as well, so if you MUST breed your dog, look for an individual that has one or two common ancestors a couple of generations back. Breeding is not to be entered into lightly. Too many people do that, and that is why so many dogs today live with genetically transmitted conditions like epilepsy, thyroid problems, hip dysplasia, von Willebrand's disease, and other conditions that end up costing owners money and dogs their health and happiness.
Again, my apologies if I offended you with my statement re: inbreeding, but I just cannot stress enough that breeding is something that way too many people do without learning everything they can about it first, without proving their dogs meet their respective breed standards first, and without putting their dogs through extensive testing to be sure they are free of genetically-passed diseases before they breed them. I have spent many years doing rescue and trying to help the already-out-of-control homeless pet problem, so this is a subject that I am passionate about. My statement is not just to those who are considering breeding, but also to those who are considering purchasing puppies from people who breed indescriminately.
Ghilly
Breeding a dog with his own pup is too close genetically for the litter to be safe, healthy and strong. Breeding 'cousins' would probably be a better answer.
I would also recommend against inbreeding your dogs. If inbreeding causes genetic defects in humans, it should do the same in dogs.
In breeding is a known and common cause of congenital disease and defects, many of which we have see here in these forums. You might talk to those people who have had to return pups or euthanize dogs from problems caused by inbreeding.
I would never recommend it. AKC won't condemn it because so many of their members do it - hence the high rate of congenital disease in inbreed dogs.
I googled your question and came up with this. Hope this helps.
AKC Fact Inbreeding
The relationship between the Sire and Dam of the litter has no bearing on registration eligibility. Anyone who considers breeding their dog must consider temperament, health, and finding mates that complement each other. Breeding dogs that are closely related can fix certain characteristics, both desirable and undesirable. Therefore, we would expect that only very experienced breeders would consider inbreeding and line breeding. However, there is no rule that prohibits inbreeding and line breeding; this is left to the discretion of individual breeders.
Inbreeding - The mating of two closely related dogs of the same breed.
Line breeding - The practice of mating a dog to a member of an earlier generation of the dog's bloodline.