Nonassociative Product

The number of nonassociative $n$-products with $k$ elements preceding the rightmost left parameter is

$$F(n,k) = F(n-1,k)+F(n-1,k-1)$$

$$= {n+k-2\choose k}-{n+k-1\choose k-1},$$

where ${n\choose k}$ is a Binomial Coefficient. The number of $n$-products in a nonassociative algebra is

$$F(n)=\sum_{j=0}^{n-2} F(n,j)={(2n-2)!\over n!(n-1)!}.$$

References

Niven, I. M. Mathematics of Choice: Or, How to Count Without Counting. Washington, DC: Math. Assoc. Amer., pp. 140-152, 1965.