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

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

− = 2 × . ==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.

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.

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.

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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