Touchard's formula

Statement

Let M_n denote the number of seating arrangements for n couples. (Touchard 1934) derived the formula

$$M_n = 2 \cdot n! \sum_{k=0}^n (-1)^k \frac{2n}{2n-k} {2n-k\choose k} (n-k)!.$$

Much subsequent work has gone into alternative proofs for this formula and into various generalized versions of the problem.

A different umbral formula for M_n involving Chebyshev polynomials of first kind was given by (Wyman Moser 1958).