Can you charge after moving at the double?

Yep, nothing prevents it.

At the double specifically forbids shooting, but says nothing about charges.

what about the 'expert riders' ability? it says something different in the back of the book, before the index? it says you can move,shoot and charge (unless attempted at the double)??

Normally shooting prevents you from charging that turn, Expert rider allows you to shoot even if you moved full distance and allows you to charge after shooting, but does not allow you to shoot if you attempt to move AtD.

The skill summaries on the quick reference sheets in the back of the book are just that, summaries. Read the full rules for Expert Rider and Skilled Rider on page 62.