The sum in particular is: $$\sum_{k=1}^n (k^2+k^k)$$
I tried looking for a closed form for $\sum_{k=1}^{n} k^k$ but from here I'm certain it doesn't exist. Even though maybe a closed form for the other one does exist. And the method I'm suggested is method of differences, which I tried factoring or try looking for that difference as the sum of squares but those are the only ideas that came to me.