History
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described above.
A rigorous proof of these formulas and his assertion that such formulas would exist for all odd powers took until Carl Jacobi in 1834 (Jacobi 1834).
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