notation - Is there a standard name or shorthand for "plustorial"?

We're all familiar with factorial:

$$n>0,\quad n! = n \times (n-1) \times \cdots \times (n-(n-1))$$

I've occasionally seen "plustorial":

$$n>0,\quad n(\mathrm{plustorial}) = n + (n-1) + \ldots + (n-(n-1))$$

Some quick web searching indicates that there is some non-standard but somewhat common usage of the term "plustorial" to describe this, with shorthand being a double-dagger or an exclaimation point having a "+" rather than a dot beneath the vertical mark.

My question is: Is there a "real" standard name for this process and is there a standard corresponding shorthand? I understand that it could be written in sigma notation, was curious about something more terse.