Jacques Touchard

Jacques André Charles Touchard (1885–1968) was a French mathematician. In 1953, he proved that an odd perfect number must be of the form ${\displaystyle 12k+1}$ or ${\displaystyle 36k+9}$. 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.

The following algebraic identity involving the Catalan numbers

${\displaystyle C_{k}={1 \over {k+1}}{{2k} \choose {k}},\quad k\geq 0}$

is apparently due to Touchard (according to Richard P. Stanley, who mentions it in his panorama article "Exercises on Catalan and Related Numbers" giving an overwhelming plenitude of different definitions for the Catalan numbers). For ${\displaystyle n\geq 0}$ one has

${\displaystyle C_{n+1}=\sum _{k\,\leq \,n/2}2^{n-2k}{n \choose 2k}C_{k}.\,}$

Using the generating function

${\displaystyle C(t)=\sum _{n\geq 0}C_{n}t^{n}={{1-{\sqrt {1-4t}}} \over {2t}}}$

it can be proved by algebraic manipulations of generating series that Touchard's identity is equivalent to the functional equation

${\displaystyle {t \over {1-2t}}C\left({t^{2} \over (1-2t)^{2}}\right)=C(t)-1}$

satisfied by the Catalan generating series ${\displaystyle C(t)}$.