We know that $$f(n)=\sum_{i=0}^n\binom{n}{i}=2^n$$ and $$g(n)=\sum_{i=0}^ni\binom{n}{i}=n2^{n-1}.$$
Are there any finite natural sums that lead to $2^{2^n}$ or $2^n2^{2^{n-1}}$ results other than $f(2^n)$ and $g(2^n)$ type ones?
In general, I do not want results that involve composition of functions that lead to $2^{2^n}$ results. For instance, if $h(n)=2^n$ then $h(2^n)=2^{2^n}$ is obvious and I want to avoid such trivialities.
If the summation remains over $n^t$ terms for fixed $t\in\Bbb N$, that is an added bonus.
Well there is this:
$\sum_{k=0}^{N-1} 2^k = \frac{2^N -1}{2-1} = 2^N -1$
So if you set : $h(N) = 1 + \sum_{k=0}^{N-1} 2^k$ you get:
$h(2^n) = 1 + \sum_{k=0}^{2^n-1} 2^k = 2^{2^n}$
Is that what you're looking for?