Count of Commutative Binary Operations with Identity/Sequence

Theorem
Let $S$ be a set whose cardinality is $n$.

Let $N$ denote the number of different commutative binary operations which have an identity element that can be applied to $S$:


 * $N = n^{\frac {n \paren {n - 1} } 2 + 1}$

The sequence of $N$ for each $n$ begins:

$\begin{array} {c|cr} n & \dfrac {n \paren {n - 1} } 2 + 1 & n^{\frac {n \paren {n - 1} } 2 + 1} \\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 4 & 81 \\ 4 & 7 & 16 \ 384 \\ 5 & 11 & 48 \ 828 \ 125 \\ \end{array}$