These where the interesting results:
A trivial sequence would be reffered to here as a sequence with constant results, i.e.,
A derivative is a sequence obtained from subtracting terms, i.e.
1)A polynomial sequence of degree n has a trivial sequence after exactly n derivatives (and none before).
2)If a sequence is trivial after n derivatives (and none before) then it is definable in a polynomial of degree n.
3)Given the sequence, (called cyclotonomic sequence of degree n)
Then its trivial sequence is n! (factorial).
4)Given a polynomial its trivial sequence is, .
There are more facts but that should suffice.
Consider the sequence,
Its derivative is,
A polynomial sequence of degree .
Therefore the sequence above it is definable as a polynomial sequence of (because the derivative reduced the exponent).
Furthermore, the trivial sequence is .
Which is why your "generalized exponent summation" can be expressed as,
These coefficient follow something called Bernoulli numbers, search for them.