Jacques Touchard
Jacques André Charles Touchard was a French mathematician. In 1953, he proved that an odd perfect number must be of the form or. In combinatorics and probability theory, he introduced the Touchard polynomials. He is also known for his solution to the ménage problem of counting seating arrangements in which men and women alternate and are not seated next to their spouses.
Touchard's Catalan identity
The following algebraic identity involving the Catalan numbersis apparently due to Touchard.
For one has
Using the generating function
it can be proved by algebraic manipulations of generating series that Touchard's identity is
equivalent to the functional equation
satisfied by the Catalan generating series.