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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.

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  • $\begingroup$ I don't even know how to do it for $k^k$; is there a question on Math.SE about that one (may be helpful)? $\endgroup$ – Zubin Mukerjee Dec 21 '14 at 12:06
  • $\begingroup$ Not every sum has explicit form. $\endgroup$ – Jihad Dec 21 '14 at 12:13
  • $\begingroup$ Are you implying that this particular sum is necessarily not summable ? $\endgroup$ – Ahmed Elyamani Dec 21 '14 at 12:15
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    $\begingroup$ @user29619: that a series is not summable or that it does not have a nice closed form expression are two completely different things. $\endgroup$ – Jack D'Aurizio Dec 21 '14 at 12:23
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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.

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