Editing Bernoulli number (section)

==A combinatorial view: alternating permutations==
{{main|Alternating permutations}}

Around 1880, three years after the publication of Seidel's algorithm, [[Désiré André]] proved a now classic result of combinatorial analysis.{{r|André1879|André1881}} Looking at the first terms of the Taylor expansion of the [[trigonometric functions]]
{{math|tan ''x''}} and {{math|sec ''x''}} André made a startling discovery.

:<math>\begin{align}
 \tan x &= x + \frac{2x^3}{3!} + \frac{16x^5}{5!} + \frac{272x^7}{7!} + \frac{7936x^9}{9!} + \cdots\\[6pt]
 \sec x &= 1 + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} + \frac{50521x^{10}}{10!} + \cdots
\end{align}</math>

The coefficients are the [[Euler number]]s of odd and even index, respectively. In consequence the ordinary expansion of {{math|tan ''x'' + sec ''x''}} has as coefficients the rational numbers {{math|''S''<sub>''n''</sub>}}.

: <math> \tan x + \sec x = 1 + x + \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 + \tfrac{5}{24}x^4 + \tfrac{2}{15}x^5 + \tfrac{61}{720}x^6 + \cdots </math>

André then succeeded by means of a recurrence argument to show that the [[alternating permutation]]s of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).