Define $F(n,i)=\prod_{j=1}^nj^{j^i}$. $F(n,0)=n!$. $F(n,1)$ is hyperfactorial.

Is there a term for $F(n,i)$?

How fast do these grow?

Is growth rate $2^{\frac{c_in^{i+1}\log_2n}{i+1}}$ with some constant $c_i>0$ (based on comments below)?

Is there a relation to special function at every $i$ (just like we have relation for factorial and hyperfactorial)?