In The Godfather, why isn't Fredo at Vito Corleone's funeral?
Well, at the time of Don Vito Corleone's death by heart failure, Fredo had been sent off to Las Vegas to begin the process of securing the family real estate and business(es) as well as ensuring that local politicians and senior police are either bought, or otherwise neutered. It is the new Godfather's decision that the family should move to Vegas and away from the troubles (other families and drugs) of New York.
As we all know, the real reason Fredo was sent, was because he was the least missed of all of the family, being the weakling, both in body and mind.
As to why he wasn't at his father's funeral, well the number one reason had to be simple: Michael didn't want him there. He had previously shown himself a coward and weakling (again) in his fumbling the job of protecting his own father during the previous attempted assassination. Not only did he miss spotting the assassins, he fumbled his gun AND just sat there crying.
Michael had already been alerted that he needed to be wary and watchful at the funeral, to spot which of his men seemed chummy with the enemy, the Barzini's, and in fact it was Tessio who brokered the (fake) peace attempt, so Michael knew not to trust Tessio from that point forward.
If Michael had allowed Fredo to attend, it would have been more likely that Fredo would have created a scene, emotionally, and embarrass the family, and more importantly, ruin Michael's opportunity to catch out the REAL traitor, Tessio.
At that point, Michael didn't know the extent of Fredo's issues at being 'left out' and that he was becoming what would be an even worse traitor.
So, seeing that both of the other answers insist on Fredo actually being at the funeral, I went back and watched the funeral/burial scene again, frame by frame. No where is Fredo present in those scenes.
In the book, Fredo is listed, by name, as being at the wake. "and of course, the Don's sons" was the phrase used for the funeral. And the burial was only a few paragraphs, with no mention of Fredo. Of course, that doesn't mean that Puzo expected him not to attend, just that he wasn't integral to the event.
As I recall, Fredo was present at the funeral in the novel, and the film was faithful enough so that a deviation like that would have been at least explained, if not taken advantage of for narrative or dramatic purposes. In other words, Coppola and Puzo covered too many bases to want the audience to puzzle over this.
Judging by the gravity and formality of Mama Corleone's funeral in the sequel, it is also extremely unlikely that Fredo would have been banned from his father's funeral at that point in the brothers' relationship.
Coppola has been quoted as saying that the funeral scene was shortened by a day in the shooting schedule and had to be rushed. The most likely explanation for Fredo's absence is that John Cazale simply wasn't available that one day.
Fredo was at the funeral. You can see him getting out of one of the line of limosines.
When you come to that scene, in the foreground you see several Mafiosi being helped out of their limos. Once they clear the camera, look to the far limos. It happens quickly, but you'll see a women (Mama) dressed in black with a large headdress being escorted by Tom Hagen. Right behind them is Fredo.