This is a continuation of this entry which derives the general solution for the coefficients of:

(5) $\displaystyle \sum_{i=1}^{n} i^k= a_{k+1}n^{k+1} + a_{k}n^{k} + a_{k-1}n^{k-1} + \dots + a_{2}n^{2} + a_{1}n$

Computed coefficients for $0 \leq k \leq 15$ were provided here.

It was noted that:

$\qquad a_{k+1}=\frac{1}{k+1}, a_k=\frac{1}{2} (k>0), a_{k-1}=\frac{k}{12} (k>1), a_{k-2} = 0 (k>2)$

These values were determined by hand. However, it’s easy for LISP code to create the symbolic expression for each coefficient. The prefix expressions used by LISP (e.g. “(+ 3 (* 4 5) 6)” ) were converted to infix form (e.g. “(3 + (4 * 5) + 6)” ) and given to Maxima for simplification. The results are here.

What’s interesting is that the coefficients:

$\qquad a_{k-2}, a_{k-4}, a_{k-6}, \dots, a_{k-24}$

are all zero.

Now I have to figure out why...