My Artscroll siddur says that the beracha Me'ein Sheva is omitted when the first night of Pesach falls on Shabbat. I have seen this in other siddurim as well. What is the reason for not saying the beracha Me'ein Sheva on the first night of Pesach?
The Shulchan Arukh (OC 487:1) rules this way, and the Magen Avraham there notes the reason is that the first night of Pesach is a Leil Shimmurim (cf. Exodus 12:42) ("a guarded night") and Mei'ein Sheva was only enacted because of dangers at night (so that no one would take too long praying and leave by himself and be harmed), but there is no need to worry about that on this night.
א"ר יוסי בר' נהיגין תמן במקום שאין יין שליח ציבור עובר לפני התיבה ואומר ברכה אחת מעין שבע וחותם במקדש ישראל ואת יום השבת
Said R. Yosé b. Rabbi, “This is the custom [in Babylonia] in a place where they have no wine [available for the recitation of the Prayer of Sanctification]. The leader goes before the ark [to recite the Prayer on Friday eve] and he recites the [special Sabbath] blessing which embodies an abstract of the seven [intermediate blessings of the daily Prayer of Eighteen] and he concludes it by saying, ‘Blessed art thou O Lord, who sanctifies Israel and the Sabbath Day.’” [Through this recitation, they fulfill their obligation to recite the Prayer of Sanctification.]
(translation from Neusner)
From this source, it seems that the recitation of the beracha me'ein sheva is there so that those who don't have wine and also cannot pray ma'ariv themselves can fulfill their obligation of kiddush hayom via the shaliach tsibbur. (This disagrees with the Bavli's reasoning on Shabbat 24b, referenced by DoubleAA in another answer.)
On Pesach evening, when everyone is obligated to have four cups of wine, and hence make kiddush on wine, it would be unnecessary to have the bracha me'ein sheva. Indeed, Guggenheimer's commentary on Brachot makes this statement:
Today, the benediction is recited always except the first night of Passover when everybody is required to drink four cups of wine; thus, there is no reason to replace Qiddush by public prayer.