Baruch Sh'Ptarani : A Parents' Blessing

(י) וַיִּגְדְּלוּ הַנְּעָרִים (בראשית כה, כז), רַבִּי לֵוִי אָמַר מָשָׁל לַהֲדַס וְעִצְבוֹנִית שֶׁהָיוּ גְּדֵלִים זֶה עַל גַּבֵּי זֶה, וְכֵיוָן שֶׁהִגְדִּילוּ וְהִפְרִיחוּ זֶה נוֹתֵן רֵיחוֹ וְזֶה חוֹחוֹ, כָּךְ כָּל י"ג שָׁנָה שְׁנֵיהֶם הוֹלְכִים לְבֵית הַסֵּפֶר וּשְׁנֵיהֶם בָּאִים מִבֵּית הַסֵּפֶר, לְאַחַר י"ג שָׁנָה זֶה הָיָה הוֹלֵךְ לְבָתֵּי מִדְרָשׁוֹת וְזֶה הָיָה הוֹלֵךְ לְבָתֵּי עֲבוֹדַת כּוֹכָבִים. אָמַר רַבִּי אֶלְעָזָר צָרִיךְ אָדָם לְהִטָּפֵל בִּבְנוֹ עַד י"ג שָׁנָה, מִיכָּן וָאֵילָךְ צָרִיךְ שֶׁיֹּאמַר בָּרוּךְ שֶׁפְּטָרַנִּי מֵעָנְשׁוֹ שֶׁל זֶה.

Bereshit Rabbah 63:10

"And the boys grew up" (Genesis 25:27) Rabbi Levi told a story: They [Jacob and Esau] were like a myrtle and a wild rose-bush growing side by side; when they reached the age of maturity, one displayed its fragrance, and the other grew its thorns. So for 13 years both went to school. But after turning 13, one went to the study hall and the other to idolatrous shrines. Rabbi Eleazar [ben Rabbi Simeon] said: A man is responsible for his son until the age of 13: thereafter he must say, "Blessed is God who has now freed me from the responsibility of this boy."

“It is written in Genesis Rabbah…that he whose son reaches the age of 13 must say the blessing, ‘Blessed is the One who has now freed me from the responsibility of this boy.’ There are those who say it the first time that the boy receives his aliyah to read the Torah. The [eighth-century] Gaon Rabbi Yehudai rose in the synagogue and said this blessing the first time that his son read the Torah.”
-Aaron ben Jacob Hakohen, 14th century Provence

Because this blessing is mentioned not in the Talmud, but only in the midrashic literature, which is held to be of secondary legal importance, there is some discussion about whether or not it has legal standing. Hence the question is asked whether or not the full blessing formula that includes the name of God (Barukh atah adonai, eloheinu melekh ha ‘olam) should be used, or whether an abbreviated version of the formula without this formal address should suffice.
The first reference to this question appears in Moses Isserles’ comment (16th-century Poland), “It is good to say [the blessing] without the [formula containing God’s] name and sovereignty.…” The question regarding its legality may actually have arisen because of a larger ambivalence. Originally just an age-of-majority status, bar mitzvah had become a ritual ceremony, which in Isserles’ time was likely a recent innovation. Since people were now saying the blessing in a public ritual setting, authorities worried about its propriety.
-Dr. Debra R. Blank, https://www.myjewishlearning.com/article/the-parents-blessing-baruch-she-ptarani/

“Nowadays it is customary to say the blessing at the point at which the youth prays or reads on the first Sabbath [after his 13th birthday] such that it will be publicly known that he is a bar mitzvah. In such a case, the father says, “Blessed is the One who has now freed me from the responsibility of this boy... up until now, the father has been punished when the son sinned because he had not educated him [as it is assumed that it is the father’s responsibility to teach the child]. The explanation is that until now the father was punished when the son sinned because he did not educate his son... The Levush explains the opposite - until now the son was punished for the father's sins. ”
-Abraham Gumbiner, Magen Avraham 225:5, 17th century Poland