http://upload.wikimedia.org/wikipedia/commons/7/74/JakobBernoulliSummaePotestatum.png
Jakob Bernoulli's famous Summae Potestatum (or Sums of Powers formula).

The Bernoulli´s Numbers were always part of the early computation of sums of integer powers, which has been the interest of mathematicians during centuries. The various methods to calculate the sum of the first n positive integers, the sum of the squares and the cubes of the first n positive integers were known for quite long, but there were no real formulas, only descriptions given by words.

Some of the great mathematicians that considered the problem were Pythagoras, Archimedes, Aryabhata or Abu Bakr al-Karaji from 500 BC up to 1.000 AD. It was not until the XVI century that any noticeable progress was made. In the west, mathematicians like Harriot, Faulhaber or Pascal played an important part on its development.

Jacob Bernoulli was the first to come up with the existence of a single sequence of constants B0, B1, B2, ... which provided a uniform formula for all sums of powers. The sums of powers formula from Bernoulli is the most useful and common formulation. These coefficients in the formula are known today as Bernoulli Numbers.

(If one needs to find a proper explanation to calculate a Bernoulli Number, best is to follow the link below to understand its extense variables and complexity)...
Source(s):
http://en.wikipedia.org/wiki/Bernoulli_number

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