In a rare 1631 work entitled Academiae Algebrae, J. Faulhaber published a number of formulae for power sums of the first positive integers. A detailed analysis of Faulhaber's work may be found in Knuth (1993) and, with a few amendments, in Knuth (2001).
Among the results presented by Faulhaber (without any indication of how they were derived) were the sums of odd powers
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where . While Faulhaber believed that analogous polynomials in with alternating signs would continue to exist for all powers , a rigorous proof was first published by Jacobi (1834; Knuth 1993).
Expressing such sums directly in terms of for powers , ..., 10 gives
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While Faulhaber was not aware of (and did not discover) Bernoulli numbers or harmonic numbers, a general formula for the sum of for from 1 to can be given in closed form by
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where is a generalized harmonic number, is the Kronecker delta, is a binomial coefficient, and is the th Bernoulli number.
In his work, Faulhaber also considered and (correctly) claimed that the -fold summation of , , ..., is a polynomial in when 3, 5, .... Additional details are given by Knuth (1993, 2001).
Any of these power sums might be termed a "Faulhaber sum."