Is it permissible for the children to take the gold and money left by their mother after her death without their father's knowledge and divide it among themselves, given that the father had told the mother the gold was hers, but then retracted after her death and claimed it was his?

It is not possible to rule on dispute cases without hearing all parties. However, generally speaking, your actions and your father's actions were wrong.

Your error lies in your monopolization of some of the money without the rest of the heirs, under the pretext that they are wealthy. This is an injustice, because they have a right to these funds, and you must re-distribute them according to Islamic law.

As for your father's error, if it is proven that the gold belonged to your mother, then he has no right to monopolize it, because it became the property of all heirs after her death. A gift transfers ownership to the recipient, and it does not revert to the giver after their death.

You must fear Allah and enable every heir to receive their right; otherwise, the matter must be brought before the Sharia court.