(12C Platinum) Sums of Powers of N numbers

[Albert Chan]

(07-29-2019 04:12 AM)Albert Chan Wrote:  Noticed a pattern with S_k(n) = Σi^k formula, when extend n to negative numbers:

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

Trivia, based on above formula, S_k(-1) = S_k(0) = 0

→ All Σi^k formulas (k positive integer), has the factor n * (n + 1)
→ All Σ(polynomial, degree > 0), has the factor n * (n + 1)

Update: just learned Σi^k formula and Bernoulli numbers are related:

For Mathematica, below formula = Sum[i^k, {k,n}], where k > 0

S[k_] := n^(k+1)/(k+1) + n^k/2 + Sum[BernoulliB[i] * Binomial[k,i] * n^(k+1-i)/(k+1-i), {i,2,k,2}]

Example: Σi^5 = n^6/6 + n^5/2 + (1/6)(10)*n^4/4 + (-1/30)(5)*n^2/2 = n^6/6 + n^5/2 + (5/12)*n^4 - n^2/12