I'd like to know the number of boolean functions on $N$ variables. There's a catch however: when a function can be obtained from another by permutating the variables, they should count only once. For example, $a\wedge b$ and $a\wedge c$ are the same (because we just swapped $b$ and $c$, assuming $N\ge 3$), but they are distinct from $\neg a\wedge b$ or $\neg(a\wedge b)$.
For $N=0,1,2$ there are $2,4,12$ such functions. I speculated the formula $$\prod_{k=0}^N\left[{N\choose k}+1\right]$$ which predicts $64$ and $700$ for $N=3,4$, but my computer simulation doesn't confirm that (counts $92$ and $8500$ respectively). However I'm sceptical about my simulation, because that sequence does not exist in OEIS.
So how many functions are there? Is my formula correct?
EDIT:
For $N=2$, with variables $a$ and $b$ we have the following functions: $true$, $false$, $a$, $\neg a$, $a\wedge b$, $\neg(a\wedge b)$, $a\vee b$, $\neg(a\vee b)$, $\neg a\wedge b$, $\neg a\vee b$, $a\otimes b$, $a\leftrightarrow b$. The following are duplicates: $b$, $\neg b$, $\neg b\wedge a$, $\neg b\vee a$. Counting those too would give the expected $2^{2^N}=16$.