Summary: The triumphal entry of Jesus into Jerusalem in Luke 19:28-40 shows us that Jesus was indeed the anticipated, coming king.
Three years ago I started preaching through The Gospel of Luke. I titled the series, “To Seek and to Save the Lost.” In the very first sermon of the series, of which this is now the 114th sermon, I said that the key to understanding Luke’s Gospel is given to us in Luke 19:10 (quickview) , where Jesus said, “For the Son of Man came to seek and to save the lost.” Jesus often identified himself as “the Son of Man,” especially in The Gospel of Luke. Jesus declared that he is the Savior who had come on a seek-and-rescue mission. And so Luke organized his entire Gospel to show who Jesus is and what he had come to do to seek and to save the lost.
One way Luke organized his Gospel is that in the first part of Jesus’ life and ministry, Jesus was seeking the lost (Luke 1:1-19:27 (quickview) ). And in the final week of Jesus’ life and ministry, Jesus was saving the lost (Luke 19:28-24:53 (quickview) ).
Or, here is another way Luke organized his Gospel:
In whatever manner Luke organized his Gospel, I want us to see that Luke 19:28 (quickview)  is the beginning of the final section of The Gospel of Luke. Luke 19:28 (quickview)  is the beginning of Jesus’ final week on earth. And it begins with his triumphal entry into the city of Jerusalem.
28 And when he had said these things, he went on ahead, going up to Jerusalem. 29 When he drew near to Bethphage and Bethany, at the mount that is called Olivet, he sent two of the disciples, 30 saying, “Go into the village in front of you, where on entering you will find a colt tied, on which no one has ever yet sat. Untie it and bring it here. 31 If anyone asks you, ‘Why are you untying it?’ you shall say this: ‘The Lord has need of it.’ ” 32 So those who were sent went away and found it just as he had told them. 33 And as they were untying the colt, its owners said to them, “Why are you untying the colt?” 34 And they said, “The Lord has need of it.” 35 And they brought it to Jesus, and throwing their cloaks on the colt, they set Jesus on it. 36 And as he rode along, they spread their cloaks on the road. 37 As he was drawing near—already on the way down the Mount of Olives—the whole multitude of his disciples began to rejoice and praise God with a loud voice for all the mighty works that they had seen, 38 saying, “Blessed is the King who comes in the name of the Lord! Peace in heaven and glory in the highest!” 39 And some of the Pharisees in the crowd said to him, “Teacher, rebuke your disciples.” 40 He answered, “I tell you, if these were silent, the very stones would cry out.” (Luke 19:28-40 (quickview) )
In Luke 9:51 (quickview)  Luke noted that Jesus “set his face to go to Jerusalem.” At that time Jesus was in Galilee and so he headed south toward Jerusalem. Since he could not pass through Samaria, he crossed east over the Jordan River into Perea, then headed south until he was opposite Jericho, where he crossed west back over the Jordan River into Judea. Luke noted several times throughout Jesus’ journey that he was on his way to Jerusalem (cf. Luke 13:22 (quickview) 17:11; 18:31; 19:11, 28). Just prior to his arrival in Jerusalem, Luke noted Jesus’ ministry in the city of Jericho, which is on the west side of the Jordan River and on the north side of the Dead Sea.