# Bernoulli number

### 1834

According to Knuth a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834.

### 1842

Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.

### 1880

&nbsp;−&nbsp; =&nbsp;2&nbsp;×&nbsp;. ==A combinatorial view: alternating permutations== Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis.

### 1893

Solving for B^{\mp{}}_m gives the recursive formulas \begin{align} B_m^{-{}} &= \delta_{m, 0} - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^{-{}}_k}{m - k + 1} \\ B_m^+ &= 1 - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^+_k}{m - k + 1}. \end{align} === Explicit definition === In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers, usually giving some reference in the older literature.

### 2002

Prior to that, Bernd Kellner computed to full precision for in December 2002 and Oleksandr Pavlyk for with Mathematica in April 2008. :* Digits is to be understood as the exponent of 10 when is written as a real number in normalized scientific notation. == Applications of the Bernoulli numbers == === Asymptotic analysis === Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula.

### 2008

Prior to that, Bernd Kellner computed to full precision for in December 2002 and Oleksandr Pavlyk for with Mathematica in April 2008. :* Digits is to be understood as the exponent of 10 when is written as a real number in normalized scientific notation. == Applications of the Bernoulli numbers == === Asymptotic analysis === Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula.

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