When I was younger I would to play around with taking differences between terms of sequences.

These where the interesting results:

A trivial sequence would be reffered to here as a sequence with constant results, i.e.,

1,1,1,1,1,....

A derivative is a sequence obtained from subtracting terms, i.e.

1,2,3,4,5,6....

Becomes,

1,1,1,.... (trivial)

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)

0^n,1^n,2^n,4^n,....

Then its trivial sequence is n! (factorial).

4)Given a polynomial

its trivial sequence is,

.

There are more facts but that should suffice.

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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,

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These coefficient follow something called Bernoulli numbers, search for them.