For a complete graph $K_{2n}$, one can easily get the number of perfect matchings (i.e. 1-factors) with the formula ${(2n)!}/({n!*2^n})$ (see the link and OEIS A001147). In graph factorizations, 1-factorizations of a complete garph $K_{2n}$ are the partitions of the edges into 1-factors.
The OEIS sequence A000438 describes the number of 1-factorizations of $K_{2n}$. It seems that counting the number of 1-factorizations of $K_{2n}$ is very difficult.
I wonder is there any formula for calculating number of 1-factorizations?