formula for summation notation involving variable powers

Question

I need help finding the formula for this summation notation: $$\sum_{k=1}^n{k^{2k} }$$ or $$1^2 + 2^4 +3^6 +.....+n^{2n} $$

And preferably not involving calculus.

Answer 1

This probably doesn't have a nice closed form, since the simpler sum of $k^k$ (called the "hypertriangular function of $n$") doesn't.

There is no OEIS entry for the sums as a sequence with increasing $n$. The OEIS entry for the related sum of $k^k$ lists a result for the ratio of consecutive terms.

If $$a_n = \sum\limits_{k=1}^{n} k^k$$

Then

$$\lim_{n\to\infty}\left(\frac{1}{n}\cdot \frac{a_{n+1}}{a_n}\right)=e$$

At best, you can hope for a similar asymptotic result for your sum.