Count of Binary Operations with Identity/Sequence

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

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


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

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

$\begin{array} {c|cr} n & \paren {n - 1}^2 + 1 & n^{\paren {n - 1}^2 + 1}\\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 5 & 243 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$