(12C Platinum) Sums of Powers of N numbers

[Albert Chan]

Noticed a pattern with S_k(n) = Σi^k formula, when extend n to negative numbers:
(see http://www.mikeraugh.org/Talks/Bernoulli...n-LACC.pdf, slide 26)

S_k(-n) = (-1)^(k+1) * S_k(n-1)

This allow the use of symmetries, to keep forward difference table numbers small.
To force 0 in the center, start i = -floor(k/2), offset = i-1

Even k example: Σi^4 formula, forward difference table, start at offset of -3 (3 numbers before 1):

16 1 0 1 16            // i^4, i = -2 to 2
-15 -1 1 15
14 2 14
-12 12
24

S₄(-3) = -S₄(2) = -(1 + 16) = -17

S₄(n) = -17 + \(16\binom{n+3}{1}-15\binom{n+3}{2}+14 \binom{n+3}{3}-12\binom{n+3}{4}+24\binom{n+3}{5}\)

Odd k example: Σi^5 formula, forward difference table, start at offset of -3 (3 numbers before 1):

-32 -1 0 1 32 243   // i^5, i = -2 to 3
31 1 1 31 211
-30 0 30 180
30 30 150
0 120
120

S₅(-3) = +S₅(2) = 1 + 32 = 33

S₅(n) = 33 - \(32\binom{n+3}{1}+31\binom{n+3}{2}-30\binom{n+3}{3}+30\binom{n+3}{4}+120\binom{n+3}{6}\)

Update: if needed, above expression can be transformed without offset.
Example: \(\binom{n+3}{6} = \binom{n}{6} + 3\binom{n}{5} + 3\binom{n}{4} +\binom{n}{3}\)         // See Vandermonde Convolution Formula