On 1/1/08, Dan Asimov <dasimov@earthlink.net> wrote:
* Fred's instantly clear induction step for a) ...
Flattery will get you nowhere --- but don't stop!
* I just read somewhere that for odd p, the (formula for the) sum of the first n pth powers is always a polynomial in the nth triangular number.
I never knew that --- why didn't I know that --- and what's worse, I've still not the foggiest notion how to go about proving it.
QUESTION: For which (p,q) is the (formula for the) sum of the first n pth powers a polynomial in (the formula for) the sum of the first n qth powers?
The elementary guess that (q+1) might divide (p+1) fails immediately when q = 2. So it doesn't look promising for any generalisation. However a proof of the case q = 1 might cast some light ... Fred Lunnon