Originally Posted by

**ThePerfectHacker** 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 $\displaystyle Ax^n+Bx^{n-1}+...+K$ its trivial sequence is, $\displaystyle A\cdot n!$.

There are more facts but that should suffice.

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Consider the sequence,

$\displaystyle 0^m,1^m+0^m,2^m+1^m+0^m,....$

Its derivative is,

$\displaystyle 1^m,2^m,3^m,....$

A polynomial sequence of degree $\displaystyle m$.

Therefore the sequence above it is definable as a polynomial sequence of $\displaystyle m+1$ (because the derivative reduced the exponent).

Furthermore, the trivial sequence is $\displaystyle m!$.

Which is why your "generalized exponent summation" can be expressed as,

$\displaystyle \frac{1}{m!}n^{m+1}+An^m+...+K$

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