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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(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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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).
The case 
is sometimes known as Nicomachus's theorem.
Expressing such sums directly in terms of 
for powers 
, ..., 10 gives
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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(17)
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(18)
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(19)
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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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(20)
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(21)
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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."
