terminology

[Joshua Zucker]

Hi folks,
suppose you have a sequence like a_n = n^3: 0, 1, 8, 27, ...

Then I call the sequence 3n^2 + 3n + 1 the "difference sequence" or
the "sequence of differences" -- is this generally accepted
terminology?  And I denote the terms Da_n.  Is there a more standard
way to write that?

And what about the differences of that?  I call it the "second
differences" and write it as D^2 a_n.

I think those terms are OK, but here's what I'm really looking for:

If you look at the sequence a_0, Da_0, D^2 a_0, D^3 a_0, ...
then in the case of n^3 you have 0, 1, 6, 6, 0, 0, 0, 0, 0, 0, ...

Is there a generally accepted terminology and/or notation for this
sequence (of the first terms of the nth differences)?

I'd like to know, both for future sequence submissions so I can use
standard terminology, and also in my role as a high school teacher.

Thanks,
--Joshua Zucker