The Rambam (Rav Moshe Maimonides, Spain-Egypt, 1135-1204) writes (Hilchot Mila, chapter 3) that when a Berit Mila is performed, the father recites not only the Beracha of "Le’hachniso Bi’brito Shel Abraham Abinu," but also the Beracha of "She’hehiyanu." The Beracha of "She’hehiyanu" is generally recited when we perform an infrequent Misva, and so at a Berit Mila, a Misva which a father, quite obviously, fulfills infrequently, it is understandable that this Beracha should be recited.
However, the Hagahot Maimoni (notes to the Rambam’s Mishneh Torah by Rav Meir of Rutenberg, 13th century) disagrees with the Rambam’s ruling, arguing that it is inappropriate to recite this joyous blessing at a Berit, when the infant is suffering pain. Although the father experiences joy over the performance of this infrequent Misva, nevertheless, as a painful wound is inflicted in the baby, the recitation of "She’hehiyanu" is inappropriate.
Rabbenu Simha questions this rationale, noting a precedent for the recitation of "She’hehiyanu" over an occasion that features both joy and pain. When a person’s father passes away, Heaven forbid, he expresses his grief and sorrow through the recitation of the Beracha, "Dayan Ha’emet," but upon receiving his inheritance, he recites "She’hehiyanu." Although the person is clearly in pain over the loss of his father, he nevertheless recites "She’hehiyanu" to express his joy over receiving an inheritance. By the same token, then, "She’hehiyanu" should be recited over the joy of the Misva of Berit Mila, notwithstanding the pain experienced by the child. Although the phrase "She’ha’simha Bi’m’ono" cannot be recited as part of Birkat Ha’mazon at a Berit, because the word "Simha" would be inappropriate in light of the child’s pain, this does not mean that "She’hehiyanu" cannot be recited.
In defense of the Hagahot Maimoni, one could argue that in the case of the passing of one’s father, the grief and the joy stem from different events – the parent’s death, and the receiving of an inheritance. When it comes to Berit Mila, however, the Beracha of "She’hehiyanu" is recited over the circumcision, which is itself a painful experience for the infant. Here, perhaps, we cannot distinguish between the joy and the sorrow, as both directly stem from the actual Berit Mila.
Regardless, Halacha follows the Rambam’s ruling that the father recites the Beracha of "She’hehiyanu" at his son’s Berit.
Summary: At a Berit Mila, the infant’s father recites both the special Beracha of "Le’hachniso Bi’brito Shel Abraham Abinu," as well as the Beracha of "She’hehiyanu."