Faulhaber's Formula

Faulhaber's formula expresses the sum of the p-th powers of the first n positive integers, 1 to the p plus 2 to the p and so on up to n to the p, as a polynomial in n of degree p plus 1. The coefficients of that polynomial are given by binomial coefficients together with the Bernoulli numbers. For p equal to 0 the sum is simply n, for p equal to 1 it is n times (n+1) divided by 2, and for p equal to 2 it is n times (n+1) times (2n+1) divided by 6. The formula is named after the early seventeenth century mathematician Johann Faulhaber, who worked out the first seventeen cases and recognized the pattern, though the special properties of its coefficients, now called Bernoulli numbers, were identified later by Jacob Bernoulli in 1713. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/