How Power Sums Give Birth to Bernoulli Numbers: A Diagonal View of an Old Problem

Every calculus student knows formulas for 1+2+...+n and 1^2+2^2+...+n^2. But where do the mysterious fractions 1/6, -1/30, 1/42, ... come from? We show that the Bernoulli numbers---long regarded as exotic objects defined by generating functions---emerge inevitably when power sum coefficients are examined through the right lens. By organizing coefficients into a two-dimensional array and scanning along diagonals rather than rows, one discovers that each diagonal carries a single constant multiplier independent of the exponent. The recurrence satisfied by these constants is, up to a sign, exactly the defining recurrence of the Bernoulli numbers. The only tools required are the binomial theorem, telescoping sums, and mathematical induction.