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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
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    As a fun side note, we have the Exponential factorial: $$a_5=5^{4^{3^{2^1}}}$$ and beyond that we have the Hyperfactorial array notation. – Simply Beautiful Art May 27 '17 at 02:02

4 Answers4

69

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. 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. – akinuri Apr 21 '18 at 20:11
  • @akinuri Photobucket make the image blurry. – Sir Intellegence-BrainStormexe 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 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
59

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
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    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
  • This one fitted perfect for py3 `N = int(input()) for i in range(1,N+1): sum = ((i**2)+i)/2

    print(int(sum))`

     – MichaelR Jun 08 '22 at 09:46
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We should also note that the factorial function has a similar look to it as the sigma summation notation; as $$\frac{n(n+1)}{2}=1+2+3+...+n=\sum_{k=1}^nk$$ $$n!=1 \cdot 2 \cdot 3 \cdot ... \cdot n=\prod_{k=1}^nk$$

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

user112167
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