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In 2012-2013, I (with experimentation and hard work) founded this image to iteratively compute the formulas for the sum of powers. In 2015, I uploaded this three part video series on it. (In the first video, I show how to draw my "Adjusted Pascal's Triangle" and how to extract the power sum formulas from it. In the second video, I abstract that image into an abstract formula/algorithm. In the third video, I provide a complete direct algebra-based proof of the formula.)

I have two (three?) questions:

  1. Has this been done before? (Is this image original by my hand, or can someone point me to some older literature on the subject?)
  2. After I saw this page on the MAA site, I asked them (this was a few years ago) the question #1 above, and I did not hear back from them at all. So my second question is, is this result (specifically the image/triangle) "trivial" or something that they didn't care to even write me back? I guess this would lead me to then ask a third question:
  3. If it is trivial, why on earth is this method/procedure not taught in school to those who are exposed to these formulas in a precalculus course? Why have I never heard of this before (even if it was "extra information" rather than being test material)?

Yes, I know of Faulhaber's formula, but I'm pretty sure that the Bernoulli numbers are a little too advanced for the average high school student, and thus I can understand why it's not mentioned. But I didn't hear about this in college either. My proof contains no advanced mathematical concepts (or induction, for that matter) so it could be shown in a college precalculus course for sure.

Speaking of which, I also constructed a "Bernoulli Number Triangle" of a similar nature. But for those who are planning to watch (or have watched the third video by the time you are reading this) my video series which contains my direct proof of my "Adjusted Pascal's Triangle", I never proved it.

I'm speechless as to why all I seem to get from academia is silence with regard to this topic. Not that it counts as acadamia, but I also posted this in /r/Math on Reddit a few years ago, and I got zero comments/responses. Am I missing something?

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    $\begingroup$ The formula seems to me like a really good work; I am only an 3rd year undergrad but I haven't heard of this. Perhaps it's a good idea to talk to someone, say a professor at a university, and publish the result? I am sure there are numerous other organizations other than MAA that can answer your stated questions. Best of luck for everything! and thank you for this knowledge! $\endgroup$ – Cute Brownie Apr 6 at 7:32
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    $\begingroup$ I'm not surprised at all this isn't taught in school, school's don't generally even teach Faulhaber's formula for low values of $n$. Sums of powers are not really the sort of thing that make it onto the curriculum. $\endgroup$ – Jack M Apr 6 at 7:45
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    $\begingroup$ Your approach is nice, and occasionally useful, But that is the problem. There are times when one needs to calculate the sums of powers, but not that often. Those sums do not arise naturally in nearly as many situations as the binomial coefficients do. So a neat way of calculating them just isn't as useful. You trot out the calculation on those few times it helps, but generally, you relegate it to reference books rather than memorizing it. I'm sorry, but the reason you are finding silence is that nobody is particularly excited about this. It is nice, but it is also niche. $\endgroup$ – Paul Sinclair Apr 6 at 16:36

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