Bernoulli Number

The Bernoulli numbers are a sequence of rational numbers that occur frequently in analysis, appearing in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler-Maclaurin formula, and in expressions for certain values of the Riemann zeta function. Two conventions for indexing the numbers appear in the literature, differing only at the first term. For every odd n greater than one the Bernoulli number is zero, and for every even n greater than zero it is negative when n is divisible by four and positive otherwise. The numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by the Japanese mathematician Seki Takakazu, whose discovery was published posthumously in 1712, a year before Bernoulli's own posthumous publication in his Ars Conjectandi of 1713. Ada Lovelace's 1842 note on the Analytical Engine describes an algorithm for generating Bernoulli numbers with Babbage's machine, making the Bernoulli numbers the subject of the first published complex computer program, though it remains disputed whether Lovelace or Babbage developed the algorithm.