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This proof demonstrates the symmetry of power sums in polynomials for even and odd degrees, highlighting mathematical structure.
It is well known that the power sums 1ᵏ⁻¹+2ᵏ⁻¹+⋯+nᵏ⁻¹ are polynomials pₖ(n) of degree k in n . We provide an elementary proof of the fact that the polynomial pₖ(x) is symmetric with respect to the line x=-1/2 or the point (-1/2,0) , depending on whether k is even or odd, respectively.
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Helmut Länger (2025) studied this question.
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