I ran across the phrase in this question, and a Google search just keeps bringing me to the Ten Commandments example found in the link on the question. I'm not finding a definition anywhere. So what is a "Catechetical Formula"?
A Catechetical Formula is reliable or time-tested or traditional way of explaining some aspect of faith. For example, when you see someone quoting the Catechism of the Catholic Church on this site, they are reciting what amounts to a catechetical formula.
Doing this is called Catechesis:
Catechesis is the act of handing on the Word of God intended to inform the faith community and candidates for initiation into the Church about the teachings of Christ, transmitted by the Apostles to the Church.
Catechesis also involves the lifelong effort of forming people into witnesses to Christ and opening their hearts to the spiritual transformation given by the Holy Spirit.
A catechism is the name given to a written work that contains a summary of all the beliefs of the faith that is used as a teaching tool.
Some catechesis is aimed at younger children, some is aimed at adults in RCIA. The entire Baltimore Catechism is a comprehensive listing of such formulas. It keeps the questions and answers simple, so that even young children can understand. Here is what one such "formula" looks like:
Q. 340. Was Jesus Christ always man?
A. Jesus Christ was not always man, but became man at the time of His Incarnation.
Many of the people visiting this site are looking for catechetical formulas: ways of explaining an aspect of their faith to someone else. For example, when Peter asked Why is Good Friday called Good? ("Is there a very obvious way to explain to someone..."), he was looking for a catechetical formula. In my answer, I recited one formula, and then tried to develop two more - the first was adjustable and aimed at a co-worker, the second one was simple, something a child might readily remember to say to a classmate.
I suspect the term does not have special meaning but is nearly a synonym for "didactic formula", i.e., an established 'recipe' for expressing a teaching (for a catechism, in the form of an answer). Just looking at the definitions of the separate words seems adequate to discern the meaning--from Google:
catechism: "A summary of the principles of Christian religion in the form of questions and answers, used for the instruction of Christians."
From The Free Dictionary:
formula: "An established form of words or symbols for use in a ceremony or procedure."
A non-Ten Commandments use was found here:
But for the Christian the resurrection has a still richer meaning. St. Paul, clearly repeating a catechetical formula that he had learned, says to the Corinthians:
(then quoting 1 Cor. 15:3-4).