Gemara

Question (Reish Lakish): The Mishnah says that in a permitted marriage, the child follows the father. A convert may marry a Mamzeres, but the children are like the blemished one (the mother)!

(Beraisa - R. Yosi): If a convert married a Mamzeres, the child is a Mamzer.

Answer #1 (R. Yochanan): The Mishnah is like R. Yehudah, who forbids a convert to marry a Mamzeres.

Answer #2: Really, the Mishnah is like R. Yosi (who permits Mamzerim to converts). 'This is' excludes cases not listed explicitly.

72b (Beraisa - R. Yosi): A convert may marry a Mamzeres;

R. Yehudah forbids.

The Halachah is, a convert is permitted to a Mamzeres, like R. Yosi.

75a (Beraisa): Ten generations of converts may marry Mamzerim. After this, it is forbidden (lest people forget that he descends from converts);

Some permit until people (actually) forget that he comes from converts.

Rishonim

Rif and Rosh (4:5): The Halachah follows R. Yosi, who permits converts to marry Mamzerim.

Rosh: This is only until 10 generations. Afterwards it is forbidden, for people forget that he is a convert and think that a Yisrael marries a Mamzeres. A Beraisa permits 10 generations of converts to may marry Mamzerim.

Rambam (Hilchos Isurei Bi'ah 15:7): A Mamzer may marry a convert. A Mamzeres may marry a convert. In both cases, the children are Mamzerim. A Mamzer may not enter Kehal Hash-m; converts are not called Kehal Hash-m.

Rambam (8): If two converts married, even though the child was conceived and born in Kedushah, he may marry a Mamzeres. The same applies to his great grandson, until it is not known that he is a convert. Afterwards, he is forbidden to a Mamzeres.

Rambam (9): If a Yisrael or Yisraelis married a convert, the child is a Yisrael in every respect, and is forbidden to a Mamzeres.

Magid Mishneh: A convert's child from a Yisraelis is no worse than a Nochri's child from a Yisraelis. The child is a full Yisrael every respect, and is forbidden to a Mamzeres. If a Yisrael marries a convert, the marriage is permitted, so the child follows his father, and is forbidden to a Mamzeres.

Rebuttal (Ran DH Ger): A Nochri's child from a Yisraelis is a full Yisrael because his lineage cannot follow his father. "L'Mishpechosam l'Veis Avosam" applies to Yisrael. When the father is a Yisrael (convert), the marriage is permitted, so the child should follow his father's lineage, and be permitted to a Mamzeres! This requires investigation.

Defense #1 (Taz EH 4:13): The child's father was initially a Nochri (Tom). Had Tom not converted, his son (Levi) from a Yisraelis would be a full Yisrael, and forbidden to a Mamzeres. If you want to say that the conversion uprooted Levi's Nochri ancestry, he is a full Yisrael, and is still forbidden to a Mamzeres. Granted, Tom may marry a Mamzeres, for he did not receive this Kedushah (to be forbidden). Regarding this Heter, it is as if he is still a Nochri. If you want to apply to Levi l'Mishpechosam l'Veis Avosam, you must consider Tom to be a full Yisrael!

Answer #2 (Beis Shmuel EH 4:37): Perhaps the rule (that in a permitted marriage, we follow the father) applies only to Yisrael, but not to converts. We hold that a convert may marry a Mamzeres, yet the child is like his mother!

Poskim

Shulchan Aruch (EH 4:22): A convert or freed slave may marry a Mamzeres. A Mamzer may marry a convert or freed slave. This is even if conception and birth were both in Kedushah, e.g. two converts married.

Beis Shmuel (35): It seems that there is no Isur of Mamzeres, but in any case one may not increase Mamzerim in Yisrael (the children will be Mamzerim).

Beis Meir: This is like Tosfos Gitin 41a (DH Lisa), who says that a half-slave may have Bi'ah with a Mamzeres, but we force his master to free him, for we do not want to increase Mamzerim. The Beis Shmuel said 'it seems', for the Ran (30a DH Gemara) forbids a slave to had Bi'ah with a Mamzeres. If so, there is no proof that Chachamim oppose increasing Mamzerim.

Shulchan Aruch (ibid.): This is only until 10 generations. After, it is forbidden, lest people forget that he is a convert, and think that a Yisrael marries a Mamzeres. The Rambam permits him to a Mamzeres, and also his great grandson, until it is not known that he is a convert. Afterwards, he is forbidden to a Mamzeres.

Shulchan Aruch (23): If a Yisrael or Yisraelis married a convert, the child is a Yisrael in every respect, and is forbidden to a Mamzeres.

Chelkas Mechokek (23): This is like the Rambam. The Ran and Tosfos (74b DH v'Harei) permit a male convert's son to a Mamzeres, for the son follows his father. The Gemara (67a) supports the Rambam. We did not ask why the Mishnah omitted a convert married to a Bas Yisrael (it is permitted and the child follows the father).

Note: According to the Rambam, why didn't the Gemara ask that this case contradicts the Klal of the Reisha!

Rejection #1 (Sha'ar ha'Melech Isurei Bi'ah 15:9): The Mishnah omitted this, for in all other cases the child is totally like the father, but our Tana permits the daughter (of a convert and Bas Yisrael) to Kehunah.

Rejection #2 (Avnei Milu'im 23 DH v'Nicha): The Ran holds that the Tana of our Mishnah is R. Eliezer ben Yakov, who says that we follow the mother's lineage. The Halachah does not follow him.

Gra (52): We find that the children are Kosher to Kehunah (77a). When the mother is a Yisraelis, this is called "mi'Kerev Achecha" (the child can be a king), and all the more so when a Yisrael marries a convert.