Editing Bernoulli number (section)

=== Sum of powers ===
{{main|Faulhaber's formula}}
Bernoulli numbers feature prominently in the [[Closed-form expression|closed form]] expression of the sum of the {{math|''m''}}th powers of the first {{math|''n''}} positive integers. For {{math|''m'', ''n'' ≥ 0}} define

:<math>S_m(n) = \sum_{k=1}^n k^m = 1^m + 2^m + \cdots + n^m. </math>

This expression can always be rewritten as a [[polynomial]] in {{math|''n''}} of degree {{math|''m'' + 1}}. The [[coefficient]]s of these polynomials are related to the Bernoulli numbers by '''Bernoulli's formula''':
: <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m \binom{m + 1}{k} B^+_k n^{m + 1 - k} = m! \sum_{k=0}^m \frac{B^+_k n^{m + 1 - k}}{k! (m+1-k)!} ,</math>
where {{math|<big><big>(</big></big>{{su|p=''m'' + 1|b=''k''|a=c}}<big><big>)</big></big>}} denotes the [[binomial coefficient]].

For example, taking {{math|''m''}} to be 1 gives the [[triangular number]]s {{math|0, 1, 3, 6, ...}} {{OEIS2C|id=A000217}}.

:<math> 1 + 2 + \cdots + n = \frac{1}{2} (B_0 n^2 + 2 B^+_1 n^1) = \tfrac12 (n^2 + n).</math>

Taking {{math|''m''}} to be 2 gives the [[square pyramidal number]]s {{math|0, 1, 5, 14, ...}} {{OEIS2C|id=A000330}}.

: <math>1^2 + 2^2 + \cdots + n^2 = \frac{1}{3} (B_0 n^3 + 3 B^+_1 n^2 + 3 B_2 n^1) = \tfrac13 \left(n^3 + \tfrac32 n^2 + \tfrac12 n\right).</math>

Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
: <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m (-1)^k \binom{m + 1}{k} B^{-{}}_k n^{m + 1 - k}.</math>

Bernoulli's formula is sometimes called [[Faulhaber's formula]] after [[Johann Faulhaber]] who also found remarkable ways to calculate [[sums of powers]].

Faulhaber's formula was generalized by V. Guo and J. Zeng to a [[q-analog|{{mvar|q}}-analog]].{{r|GuoZeng2005}}