Integral cohomology of an odd-sphere loop space ($H^{\ast }(\Omega S^{2n+1};\mathbb{Z})={\Gamma }_{\mathbb{Z}}(a_{2n})$)

For $n\ge 1$, there is one integral cohomology generator $a_k$ in every degree $2nk$, and its products are $a_i{a}_j=\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)a_{i+j}$. The Bott–Samelson theorem supplies free polynomial homology; its primitive generator gives the binomial diagonal dual to this divided power algebra.