83

Is there a notation for addition form of factorial?

$$5! = 5\times4\times3\times2\times1$$

That's pretty obvious. But I'm wondering what I'd need to use to describe

$$5+4+3+2+1$$

like the factorial $5!$ way.

EDIT: I know about the formula. I want to know if there's a short notation.

akinuri
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    $1+2+\dots+n=\dfrac{n(n+1)}{2}$; there's no need for a special notation. – egreg Dec 04 '13 at 23:28
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    Duplicate of: http://math.stackexchange.com/q/60578/439 – Niel de Beaudrap Dec 04 '13 at 23:33
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    @NieldeBeaudrap I'm asking about it's notation... – akinuri Dec 04 '13 at 23:36
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    The sigma notation is a notation for it. – user112167 Dec 04 '13 at 23:37
  • Did you read the answers in the linked post? – Niel de Beaudrap Dec 04 '13 at 23:38
  • @NieldeBeaudrap Yes, but I wanted to know if there's a simpler way like factorial. Using just a single character... In Sigma Notation, it looks like M rolled over after getting drunk and numbers are partying around it. Sorry if that sounded sarcastic, but yeah. – akinuri Dec 04 '13 at 23:47
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    So, you didn't see the answer which described that Knuth suggested the notation "$n?$" ? – Niel de Beaudrap Dec 04 '13 at 23:54
  • Also... I think that you should rather get used to seeing $\Sigma$, $\Pi$, and other Greek letters with some regularity if you are interested in mathematics. Overcoming some minor notational prejudices early will prove its own reward. – Niel de Beaudrap Dec 04 '13 at 23:56
  • @NieldeBeaudrap Actually I did. I was googling about it. I suppose $n?$ is the closest thing to what I asked for. You're right. I should. I was just working on something and spent few pages on it. It's just um.. consuming writing the same thing over and over again. Wanted an easy fix, like $n!$. I'll use $n?$ for now. Thanks for the replies. – akinuri Dec 05 '13 at 00:05
  • If I might suggest, you could always define a function for the purpose if the question marks make your math look cluttered or strange. Given the fact that these are triangular numbers, $\tau(n)$ would be appropriate and distinctive. – Niel de Beaudrap Dec 05 '13 at 00:18
  • As a fun side note, we have the [Exponential factorial](http://googology.wikia.com/wiki/Exponential_factorial): $$a_5=5^{4^{3^{2^1}}}$$ and beyond that we have the [Hyperfactorial array notation](http://googology.wikia.com/wiki/Hyperfactorial_array_notation). – Simply Beautiful Art May 27 '17 at 02:02

4 Answers4

58

It is called the $n$th triangle number and it can be written as $\binom{n+1}2$, as a binomial coefficient.

endolith
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Berci
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    why would it be called a "triangle number"? – khaverim Jun 10 '16 at 17:01
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    @khaverim Check this [image](http://i1115.photobucket.com/albums/k544/akinuri/nth%20triangle%20number-01.jpg). This is something I've come up with some time ago to visualize and understand how the calculation works. I've literally spent an hour or so to think that, becasuse I had nowhere to look then. And I'm guessing it's called triangle number(s) becase you can treat the number set as the half of a rectangle, a triangle. – akinuri Jul 18 '16 at 21:45
  • I assumed it was a triangle number due to the obvious relationship to Pascal's Triangle. – James Antill Aug 11 '16 at 21:41
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    Unless I'm misunderstanding the notation, this is not a correct answer. I believe it should be ( ( n ( n + 1 ) ) / 2 ), not ( ( n + 1 ) / 2 ). – Oliver Nicholls Aug 25 '17 at 09:37
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    Berci just skipped some details. The notation is called [binomial coefficient](https://en.wikipedia.org/wiki/Binomial_coefficient). – akinuri Apr 21 '18 at 20:11
  • @akinuri Photobucket make the image blurry. – BrainStorm.exe Sep 26 '19 at 22:21
  • @BrainStorm.exe Thanks for the notification. The image at the direct link is (now) blurry, because it seems my account exceeded the free plan bandwidth (25MB/month). This [link](https://photobucket.com/gallery/user/akinuri/media/cGF0aDovbnRoIHRyaWFuZ2xlIG51bWJlci0wMS5qcGc=) seems to work fine. – akinuri Sep 27 '19 at 07:10
  • This is incorrect. This not only does not get the desired result, it also doesn't get the $n$th triangle; it's just a simple linear function which better expresses "What is one more than the sum of $n$ halves?" – Ky. Aug 28 '20 at 15:14
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    @BenLeggiero it's correct, it's binomial coefficient notation as pointed out by other commenters, not a linear function. It could otherwise be written as C(n+1, 2), n+1C2, etc., and translates as (n+1)!/(2(n-1)!), or n(n+1)/2 – 17slim Dec 09 '20 at 20:26
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    @17slim Oh! My apologies; I thought this was a fraction. – Ky. Dec 09 '20 at 21:36
48

That can be done with the formula $\frac{n^2+n}{2}$

imranfat
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  • What about doing the opposite, finding the dimensions using the output number? So far I have `floor(sqrt(2 * s))` – Aaron Franke Sep 08 '20 at 21:57
  • If the output is, say $y$ then you need to solve $n^2+n=2y$ or $(n+0.5)^2=2y+0.25$. Take square root and hope for an integer answer – imranfat Sep 09 '20 at 13:55
20

We should also note that the factorial function has a similar look to it as the sigma summation notation; as $$\frac{n(n+1)}{2}=1+2+3+...+n=\sum_{k=1}^nk$$ $$n!=1 \cdot 2 \cdot 3 \cdot ... \cdot n=\prod_{k=1}^nk$$

Eleven-Eleven
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15

$\sum_{n=1}^{k} n = 1 +2+3+\ldots+k$. Is a nice notation for it. So $$1 + 2 + 3 + 4 + 5 = \sum_{n=1}^{5} n$$.

user112167
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