Is there a closed form expression for the "generalized" addition of the first $n$ numbers?

The recurrence can be written as:

$$ \begin{aligned} \sigma(k + 1, n) &= \sigma(k, n) + \frac{\sigma(k, n)^2 + \sigma(k, n)}{2}\\ &= \frac{\sigma(k, n)^2 + 3\sigma(k, n)}{2}\\ \end{aligned} $$

Since $n$ doesn't matter for this analysis, I'll rewrite with $\sigma(n, k) = \sigma_k$ for conciseness.

$$ \sigma_{k+1} = \frac{\sigma_k^2 + 3\sigma_k}{2} $$

This is a quadratic recurrence relation. I'll put it in standard form $S_{k+1} = aS_k^2 + bS_k + c$.

$$ \sigma_{k+1} = \frac{1}{2}\sigma_k^2 + \frac{3}{2}\sigma_k $$

Now we're just trying to find a closed form for a quadratic recurrence relation. Following Will Jagy's answer here, we first try to create another sequence $T_k$ with a recurrence of the form $T_{k+1} = T_k^2 + c$. It turns out this sequence is just:

$$ T_k = \frac{1}{2}\sigma_k + \frac{3}{4} $$

with the recurrence

$$ T_{k+1} = T_k^2 + \frac{3}{16} $$

(Check me!). Will Jagy then says that there are only two cases with a closed form: when $c = 0$ and when $c = -2$. Neither case holds, so there isn't a closed form.

I would love to know more about this problem in generality. Why are those two cases the only ones that can be solved?