Faulhaber's formula

In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ${\displaystyle p}$th powers of the first ${\displaystyle n}$ positive integers $${\displaystyle \sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}}$$ as a polynomial in ${\displaystyle n}$. In modern notation, Faulhaber's formula is $${\displaystyle \sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{r=0}^{p}{\binom {p+1}{r}}B_{r}^{+}n^{p+1-r}.}$$ Here, ${\textstyle {\binom {p+1}{r}}}$ is the binomial coefficient "${\displaystyle p+1}$ choose ${\displaystyle r}$", and the ${\displaystyle B_{r}^{+}}$ are the second Bernoulli numbers, identical to the first ones except for ${\textstyle B_{1}^{+}={\frac {1}{2}}}$.