"And when they had sung [you]an hymn, [/you]they went out into the mount of Olives." (Mat. 26:30).
More properly, "sung the hymn" is an intransitive. If it were transitive, it would be "sung a hymn" which would include an object. Like saying, "sung psalm 113".
Intransitive has no object, which would render it more generally. A clearer rendering would be "sung [you]the [/you]hymn". What would "the hymn" be to be sung at the Passover? Certainly, Christ and all the disciples sung it, and it had to be what they were acquainted with. Did the Jews know what "the hymn" would be? Yes, it was the second part of the Hallel. The Passover was observed by the Jews by singing Psalm 113-114 at the start, and then 115-118 at the close (Keep in mind the Jewish Text stands one psalm back 112-117 because of their division). It was "the" section to be sung at the close of the ordinance; "the hymn."
This 118th psalm was already quoted by Matthew in 21:9 and 21:42, "Blessed be he that cometh in the name of the LORD: we have blessed you out of the house of the LORD." (Psa. 118:26). Would that have applied to Christ's institution of the Supper? Most assuredly.
Christ intended to instruct his church in this, (for he does nothing by accident, especially a reconstituting an old ordinance into a new one) where the Lord's Supper is a service of thanksgiving and joy. Is the church instructed to act a certain way when joyful? Will Scripture furnish us through the analogy of faith as to what both Christ and his disciples sung, and why they sung it? Again, most assuredly. The Christian is to give thanks to God. Singing is an external action which has as it substance the word of God, (i.e. "Let the [you]word of Christ[/you] dwell in you richly in all wisdom," (Col. 3:16), where Christ has appointed singing to express the inward joy and thankfulness of our hearts. That is a very basic OT concept. This is why James says, "Is any merry? let him sing psalms." (Jam. 5:13). Merry means, "gladdened in the heart, joyful or cheerful in mind." Were the disciples joyful at Christ's institution of the Lord's Supper? We know they were not afflicted, because they would have [you]prayed[/you], not sung. How do we know this? "Is any among you afflicted? let him pray." (Jam. 5:13). They were not afflicted. How do we know this? They [you]sung, [/you]not prayed, at the Passover, which Christ reconstituted into a glorious sacrament of thanksgiving and joy. Jesus institutes the sacrament of thanksgiving, and so they sing. [you]What do they sing?[/you] To keep in step with not violating Scripture, if they are joyful, they sing psalms. James says that such singing surrounds singing psalms. Jesus did not make up a new song at the Passover, rather, he sung "the" hymn, the psalm, per James, the OT and the Evangelical meaning behind the word hymnos. (James's statements on praying and singing were always the case (in the OT). He does not invent something new, and Christ would have known that already.)
At the very least, every time the Lord's Supper is instituted and taken in a church, that church should end with singing a psalm, because they are merry and joyful otherwise they violate James 5:13.
Further, though, this CLEARLY shows that hymn means psalm in this passage per Jesus' use of the ordinance of singing while joyful, and James' subsequent instruction to that end, otherwise, Christ would be in violation of the direct command of God to sing psalms when joyful.
(Side note: consider Psalms 34:4, 107:6, 13, 28, etc., for afflicted prayer. Consider being merry in Psalm 28:7; 65:1; 108:1, etc.)