Faulhaber's formula expresses the sum of the p-th powers of the first n positive integers, 1 to the p plus 2 to the p and so on up to n to the p, as a polynomial in n of degree p plus 1. The coefficients of that polynomial are given by binomial coefficients together with the Bernoulli numbers. For p equal to 0 the sum is simply n, for p equal to 1 it is n times (n+1) divided by 2, and for p equal to 2 it is n times (n+1) times (2n+1) divided by 6. The formula is named after the early seventeenth century mathematician Johann Faulhaber, who worked out the first seventeen cases and recognized the pattern, though the special properties of its coefficients, now called Bernoulli numbers, were identified later by Jacob Bernoulli in 1713.
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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
Attributed To: Johann Faulhaber, Lead sentence
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} .
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Faulhaber's major contribution was in calculating the sums of powers of integers;
Wolfram MathWorld: Faulhaber's Formula
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