 These polynomials are divisible by because the Bernoulli number is 0 for odd.Faulhaber also knew that if a sum for an odd power is given by

then the sum for the even power just below is given by

Note that the polynomial in parentheses is the derivative of the polynomial above with respect to "a".Since "a" = "n"("n" + 1)/2, these formulae show that for an odd power (greater than 1), the sum is a polynomial in "n" having factors "n"2 and ("n" + 1)2, while for an even power the polynomial has factors "n", "n" + ½ and "n" + 1.