Reciting "Shehechiyanu" At a Bris

Much of the following is from Rabbi Pirutinsky's discussion of the subject in Sefer HaBris, starting page 290. He quotes all the Rishonim and Acharonim. I am making it simple through just giving the opinions. 

One of the more impactful blessings we have in our liturgy is the "Shehechiyanu"

ברוך אתה ה' א-לקינו מלך העולם שהחיינו וקימנו והגיענו לזמן הזה
Blessed are you Hashem, our God, King of the universe, Who has had us live, and survive, and brought us to this time.

We say this blessing on holidays and other special occasions, mostly when a special event has not happened for over 30 days. (There is an entire Siman in the Shulchan Arukh - Orach Chaim 225 - dedicated to this blessing!)

Do we say it at a bris?


There are three answers to this question:
1. Yes
2. No
3. It depends

YES
Those who say it are usually Sefardim or of Middle Eastern descent. It is the custom in Israel to say it, even among Ashkenazi Jews. I have seen some Ashkenazic Jews from Israel say it at their son's bris in the United States. The thought process is simple. How often do you have a son? How often do you  have a bris? Say Shehechiyanu!

NO
The reasons to not say it are a little more formal thinking.
1. The child experiences pain at the bris (צערא דינוקא), so it is not a nice time to say Shehechiyanu.
2. Unlike a Pidyon Haben, which takes place after the child's 30th day, in most cases (barring extreme medically required delays) the bris takes place while the baby is under 30 days old. In Halakhic terms, the child is still in the realm of what's called a חשש נפל. While (THANK GOD) most babies survive to 30 days, halakha assigns a status of נפל to a child under 30 days "just in case" the worst happens. Because of that concern, there is hesitation to allow the recitation of Shehechiyanu on the bris.

IT DEPENDS
A special circumstance to allow for Shehechiyanu would be if the father himself is doing the circumcision. Since in most cases the mohel is performing the mitzvah on the father's behalf, and since for the mohel it's a regular occurence, the mohel would not be saying Shehechiyanu. And since the father isn't actually doing the milah he can't say it either!
But if the father is either a mohel himself, or the mohel sets it up for the father to do, then there are opinions which support the father's saying Shehechiyanu in this circumstance.