Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in number theory. The values of the first 20 Bernoulli numbers are given in the adjacent table. For every even n other than 0, Bn is negative if n is divisible by 4 and positive otherwise. For every odd n other than 1, Bn = 0. The superscript ± used in this article designates the two sign conventions for Bernoulli numbers. Only the n = 1 term is affected: In the formulas below, one can switch from one sign convention to the other with the relation B n + = ( − 1 ) n B n − {displaystyle B_{n}^{+{}}=(-1)^{n}B_{n}^{-{}}} . The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {displaystyle B_{n}(x)} , with B n − = B n ( 0 ) {displaystyle B_{n}^{-{}}=B_{n}(0)} and B n + = B n ( 1 ) {displaystyle B_{n}^{+}=B_{n}(1)} (Weisstein 2016). Since Bn = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write 'Bn' to mean B2n. This article does not follow this notation. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712 (Selin 1997, p. 891; Smith & Mikami 1914, p. 108) in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine (Menabrea 1842, Note G). As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.