Factorial, but with addition [duplicate]

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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.

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4 Answers

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

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

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

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

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