Sources that child born to gentile father and Jewess is gentile:

This opinion is most explicit in Tosfos Kiddushin 75b (and is consistent with Tosfos' opinion in numerous other places, noteably Yevamos 15 & 23). [See Maharsha ad loc, and Piskei Tosfos]

This is also the opinion of Rashi (Kiddushin 67b and Parshas Emor, regarding the megadef). Rashi seemingly contradicts this position in Kiddushin 75a, ruling that a child fathered by an eved kenaani and a Jewess is a Jew.

See Tosfos Rid (Gittin 41b) who [like Rashi] differentiates between son of an eved & Jewess to a gentile & Jewess, based on the dictum עבד אין לו חייס. Chemdas Shlomo (E"H §2) quotes the Nesivos (also in Shu"t R' Yackov MeLisa from the author of the Nesivos) who says something very similar. (See also Ohr Someach Ch. 15 of Issurei Biah for different rationale to the same distinction).

The [Maharit Alagazi][1] suggests that the status of the child is defined by who raises him - his gentile father or his Jewish mother. While this would answer some of the apparent contradictions, the logic is difficult to understand.

Halacha:

R' Akiva Eiger (Shu"a Yo"d 266), Maharshal (cited in Drisha 266.15) and Maharam Shik (EH"E §20) writes that one should not circumcise a child of gentile father and Jewess on Shabbos, out of concern for the opinions that he is not Jewish.

R' Eliezer Gordon (Kuntres Achron, §1) contends this may be the opinion of the Rambam & Rif; see his rather lengthy discussion of the subject for a full synopsis.

IIRC (I don't have the sefer on hand), the Nesivos in (Shu't R' Yackov Melisa §26) doesn't allow the child born from union of Gentile and Jewess to marry a Jew/ess without Geirus.

Poskim who conclude child is halachically Jewish:

רמ”א (יו”ד סי’ קנט סעי’ ג) ס’ עצי ארזים (סי’ ד ס”ק ד ד”ה היוצא) ס’ ערוגות הבשם (סי’ ד ס”ק ד) פני יהושע (גיטין מ ע”א תוס’ ד”ה אותו) שיירי קרבן (גיטין פ”ד הל’ ב) תפארת למשה (יו”ד סי’ רסו) שו”ת ברית אברהם (סי’ ב אות ב) שו”ת נודע ביהודה (מהדו”ת אבהע”ז סי’ קנ) שו”ת חמדת שלמה (אבהע”ז סי’ ג) ישועות יעקב (אבהע”ז סי’ ד פי’ הארוך ס”ק ו) שו”ת בנין ציון (ח”א סי’ כא) שו”ת יכין ובועז (ח”א סי’ עה וסי’ קז, ובח”ב סי’ ג) שו”ת אחיעזר (ח”ג סי’ כא) חזון איש (אבהע”ז סי’ ו ס”ק ז).

[See also Chiddushei Grana"t §11, Imrei Moshe §26, Chiddushei R' Reuven Yevamos §22] [1]: https://en.wikipedia.org/wiki/Yom_Tov_Algazi

Sources that child born to gentile father and Jewess is gentile:

This opinion is most explicit in Tosfos Kiddushin 75b (and is consistent with Tosfos' opinion in numerous other places, noteably Yevamos 15 & 23). [See Maharsha ad loc, and Piskei Tosfos]

This is also the opinion of Rashi (Kiddushin 67b and Parshas Emor, regarding the megadef). Rashi seemingly contradicts this position in Kiddushin 75a, ruling that a child fathered by an eved kenaani and a Jewess is a Jew.

See Tosfos Rid (Gittin 41b) who [like Rashi] differentiates between son of an eved & Jewess to a gentile & Jewess, based on the dictum עבד אין לו חייס. Chemdas Shlomo (E"H §2) quotes the Nesivos (also in Shu"t R' Yackov MeLisa from the author of the Nesivos) who says something very similar. (See also Ohr Someach Ch. 15 of Issurei Biah for different rationale to the same distinction).

The Maharit Alagazi suggests that the status of the child is defined by who raises him - his gentile father or his Jewish mother. While this would answer some of the apparent contradictions, the logic is difficult to understand.

Halacha:

R' Akiva Eiger (Shu"a Yo"d 266), Maharshal (cited in Drisha 266.15) and Maharam Shik (EH"E §20) writes that one should not circumcise a child of gentile father and Jewess on Shabbos, out of concern for the opinions that he is not Jewish.

R' Eliezer Gordon (Kuntres Achron, §1) contends this may be the opinion of the Rambam & Rif; see his rather lengthy discussion of the subject for a full synopsis.

IIRC (I don't have the sefer on hand), the Nesivos in (Shu't R' Yackov Melisa §26) doesn't allow the child born from union of Gentile and Jewess to marry a Jew/ess without Geirus.

Poskim who conclude child is halachically Jewish:

רמ”א (יו”ד סי’ קנט סעי’ ג) ס’ עצי ארזים (סי’ ד ס”ק ד ד”ה היוצא) ס’ ערוגות הבשם (סי’ ד ס”ק ד) פני יהושע (גיטין מ ע”א תוס’ ד”ה אותו) שיירי קרבן (גיטין פ”ד הל’ ב) תפארת למשה (יו”ד סי’ רסו) שו”ת ברית אברהם (סי’ ב אות ב) שו”ת נודע ביהודה (מהדו”ת אבהע”ז סי’ קנ) שו”ת חמדת שלמה (אבהע”ז סי’ ג) ישועות יעקב (אבהע”ז סי’ ד פי’ הארוך ס”ק ו) שו”ת בנין ציון (ח”א סי’ כא) שו”ת יכין ובועז (ח”א סי’ עה וסי’ קז, ובח”ב סי’ ג) שו”ת אחיעזר (ח”ג סי’ כא) חזון איש (אבהע”ז סי’ ו ס”ק ז).

[See also Chiddushei Grana"t §11, Imrei Moshe §26, Chiddushei R' Reuven Yevamos §22]