Faulhaber's formula expresses a power sum of the first 
positive integers as a polynomial
in 
.
In Academia Algebrae, Faulhaber (1631) 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 (2003).
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 odd powers 
, a rigorous proof was first published
by Jacobi (1834), as discussed by 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
for a nonnegative integer 
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.
Bernoulli (1713, pp. 97-98) contrasted his general rule for computing power sums with the work Boulliau (1682) had devoted to the first six powers.
In his work, Faulhaber also considered and (correctly) claimed that, for a positive integer 
, the 
-fold sum of 
, 
, ..., 
is 
times a polynomial in 
when 
, 3, 5, .... Additional details
are given by Knuth (1993, 2003).
Any of these power sums might be termed a "Faulhaber sum."
