Faulhaber's formula
expression for sums of powers
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the
p
{\displaystyle p}
th powers of the first
n
{\displaystyle n}
positive integers
∑
k
=
1
n
k
p
=
1
p
+
2
p
+
3
p
+
⋯
+
n
p
{\displaystyle \sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}}
as a polynomial in
n
{\displaystyle n}
. In modern notation, Faulhaber's formula is
∑
k
=
1
n
k
p
=
1
p
+
1
∑
r
=
0
p
(
p
+
1
r
)
B
r
+
n
p
+
1
−
r
.
{\displaystyle \sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{r=0}^{p}{\binom {p+1}{r}}B_{r}^{+}n^{p+1-r}.}
Here,
(
p
+
1
r
)
{\textstyle {\binom {p+1}{r}}}
is the binomial coefficient "
p
+
1
{\displaystyle p+1}
choose
r
{\displaystyle r}
", and the
B
r
+
{\displaystyle B_{r}^{+}}
are the second Bernoulli numbers, identical to the first ones except for
B
1
+
=
1
2
{\textstyle B_{1}^{+}={\frac {1}{2}}}
.
The result: Faulhaber's formula
Faulhaber's formula concerns expressing the sum of the
p
{\displaystyle p}
th powers of the first
n
{\displaystyle n}
positive integers
∑
k
=
1
n
k
p
=
1
p
+
2
p
+
3
p
+
⋯
+
n
p
{\displaystyle \sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}}
as a
(
p
+
1
)
{\displaystyle (p+1)}
th-degree polynomial function of
n
{\displaystyle n}
.
The first few examples are well known. For
p
=
0
{\displaystyle p=0}
, we have
∑
k
=
1
n
k
0
=
∑
k
=
1
n
1
=
n
.
{\displaystyle \sum _{k=1}^{n}k^{0}=\sum _{k=1}^{n}1=n.}
For
p
=
1
{\displaystyle p=1}
, we have the triangular numbers
∑
k
=
1
n
k
1
=
∑
k
=
1
n
k
=
n
(
n
+
1
)
2
=
1
2
(
n
2
+
n
)
.
{\displaystyle \sum _{k=1}^{n}k^{1}=\sum _{k=1}^{n}k={\frac {n(n+1)}{2}}={\frac {1}{2}}(n^{2}+n).}
For
p
=
2
{\displaystyle p=2}
, we have the square pyramidal numbers
∑
k
=
1
n
k
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
=
1
3
(
n
3
+
3
2
n
2
+
1
2
n
)
.
{\displaystyle \sum _{k=1}^{n}k^{2}={\frac {n(n+1)(2n+1)}{6}}={\frac {1}{3}}(n^{3}+{\tfrac {3}{2}}n^{2}+{\tfrac {1}{2}}n).}
The coefficients of Faulhaber's formula in its general form involve the second Bernoulli numbers
B
j
+
{\displaystyle B_{j}^{+}}
which nearly coincide with the first Bernoulli numbers denoted
B
j
−
{\displaystyle B_{j}^{-}}
(or simply
B
j
{\displaystyle B_{j}}
); the sole exception is at
j
=
1
{\displaystyle j=1}
, where
B
1
−
=
−
1
2
{\displaystyle B_{1}^{-}=-{\tfrac {1}{2}}}
but
B
1
+
=
1
2
{\displaystyle B_{1}^{+}={\tfrac {1}{2}}}
. The Bernoulli numbers begin
B
0
=
1
B
1
+
=
1
2
B
2
=
1
6
B
3
=
0
B
4
=
−
1
30
B
5
=
0
B
6
=
1
42
B
7
=
0
,
{\displaystyle {\begin{aligned}B_{0}&=1&B_{1}^{+}&={\tfrac {1}{2}}&B_{2}&={\tfrac {1}{6}}&B_{3}&=0\\B_{4}&=-{\tfrac {1}{30}}&B_{5}&=0&B_{6}&={\tfrac {1}{42}}&B_{7}&=0,\end{aligned}}}
Then Faulhaber's formula is that
∑
k
=
1
n
k
p
=
1
p
+
1
∑
r
=
0
p
(
p
+
1
r
)
B
r
+
n
p
+
1
−
r
.
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