If we have a name and notation for the product of natural numbers up to and including n (factorial and !), then what is the name and shorthand for the sum of natural numbers up to and including n?

(I know about triangular numbers, but that's not a name for the expression that sums the corresponding series.)

  • $\begingroup$ I've heard a friend use the term "summorial" when introducing the idea to a class that had already heard about factorials. I don't like the sound of the word, but ... I'm not the judge of what's acceptable in the language. A google search for "summorial" leads to the other standard name -- it's called the "$n$th triangular number," but there doesn't seem to be a standard notation for it. $\endgroup$ Jun 7, 2015 at 17:46
  • $\begingroup$ triangular numbers? $\endgroup$
    – abel
    Jun 7, 2015 at 17:48
  • $\begingroup$ "Then what is the name and shorthand for...," "(I know about triangular numbers, but...)". Why is the $n^{th}$ triangular number (denoted $T(n)$) not an acceptable shorthand? It isn't any different than defining the function $!~:\mathbb{N}\to\mathbb{N}$ such that $!(n) = \prod_{i=1}^n i$. For shorthand, we often write $n!$ instead of $!(n)$, (just like how we write $a+b$ instead of $+(a,b)$). There is no difference mathematically though. We just have $T~:~\mathbb{N}\to\mathbb{N}$ defined as $T(n)=\sum_{i=1}^n i$ $\endgroup$
    – JMoravitz
    Jun 7, 2015 at 17:50
  • $\begingroup$ JMoravitz - I often see this sum come up in everyday life and I'm not a mathematician. I would like to be able to look at it and say "triangle of 5" or "triangle of 10" just like we can say "factorial of n" for the product case. It is this name I was looking for in OP $\endgroup$
    – Balage
    Jun 7, 2015 at 17:54
  • $\begingroup$ Again, what is it about the phrase "$n^{th}$ triangle number" or the phrase "$T(n)$" that displeases you? They both are short and easy to say, and they both have clear definitions in the context given. Remember that all operations are in fact functions themselves. If you really wanted to, you could define $nT$ as $T(n)$, and use $T$ the same way you use $!$. $\endgroup$
    – JMoravitz
    Jun 7, 2015 at 17:59

1 Answer 1


Thanks for the hint, Peter Woolfitt. Found that reference material, "The ARML Power Contest" By Thomas Kilkelly. Chapter 32 discusses the origins of the name deltorial and provides the notation $\Delta_n$ . The use of the name and notation are not mainstream, that's for sure. Does that make it any less valid?

Available on Google Books: full text


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