# What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)

I'm aware of Sigma notation, but is there a function/name for e.g.

$$4 + 3 + 2 + 1 \longrightarrow 10 ,$$

similar to $$4! = 4 \cdot 3 \cdot 2 \cdot 1 ,$$ which uses multiplication?

Edit: I found what I was looking for, but is there a name for this type of summation?

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The name for $$T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +(n-1)+n = \frac{n(n+1)}{2} = \frac{n^2+n}{2} = {n+1 \choose 2}$$

is the $n$th triangular number. Here's a picture that demonstrates the reasoning for the name:

$$T_1=1\qquad T_2=3\qquad T_3=6\qquad T_4=10\qquad T_5=15\qquad T_6=21$$

$\hskip1.7in$

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Donald Knuth in The Art of Computer Programming calls the $n$-th triangular number the "termial function", and denotes it

$$n? = 1 + 2 + ... + n = \sum_{k=1}^n k$$

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Wow, really? What volume/page does he define this termin-ology? – Niel de Beaudrap Aug 30 '11 at 1:06
Volume 1, section 1.2.5 I believe, it is in the "Permutations and Factorials" section. – tlehman Aug 30 '11 at 1:11
It's not terminal, it's termial. It also doesn't matter why he put it in his books, it is exactly what the questioner was asking about. – tlehman Aug 30 '11 at 12:38
ah, then I managed to misread it several times. Also –– if you will forgive me –– I was somewhat skeptical that Knuth would deign to give this function a name (especially when I thought that name was supposed to be "terminal", which made little sense to me); I wanted to see for myself, and also see why he would do so. – Niel de Beaudrap Aug 30 '11 at 12:41
@Niel: concerning "for pedagical reasons": I'd say, the additive analogon of a "factor" in a multiplication is "summand", so then it should rather be called "summorial" or "summatorial" – Gottfried Helms Oct 4 '11 at 6:14

Actually, I've found what I was looking for.

From the wiki on Summation:

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These numbers are also called the triangular numbers. You might think of the triangular numbers as naming a sequence: 1, 3, 6, 10, 15, 21,... But a sequence of integers is really just a function from $\mathbb{N}$ to $\mathbb{Z}$, so the triangular numbers also name the function you've written above. – Jonas Kibelbek Aug 29 '11 at 20:59

Not exactly a name, but note that $$\sum\limits_{k=1}^{n} k= \frac{n(n+1)}{2}={n+1 \choose 2}$$

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proposals for your own personal notes:

I've thought about this for years and see a clear need for a shorthand version instead of writing a little equation. Any name or symbol you invent for this purpose should be easy to remember and easy to infer it's meaning by people unfamiliar with it. Factorial should become Smacktorial because its humorous sound makes it easy to remember and the "Sm" reminds you of "sum" which tells you it's a summation version of factorial. The symbol should look like "!" with a hint of addition so it should be a plus sign with a dot underneath or alternatively described as an exclamation point with a short horizontal line through the vertical part. It's too bad you can't easily type that but if you want to use those ideas for your own personal notes, then that's my two cents on the subject.

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