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mathematical analysis > calculus > sequence > integer sequence > Euler number

mathematical analysis > combinatorics > Euler number

algebra > combinatorics > Euler number

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Euler number

In mathematics, the Euler numbers are a sequence E_n of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion

${\\displaystyle {\ rac {1}{\\cosh t}}={\ rac {2}{e^{t}+e^{-t}}}=\\sum _{n=0}^{\\infty }{\ rac {E_{n}}{n!}}\\cdot t^{n}}$ ,

where ${\\displaystyle \\cosh(t)}$ is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:

${\\displaystyle E_{n}=2^{n}E_{n}({\ frac {1}{2}}).}$

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Euler_numbers)

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RDF/XML TURTLE JSON-LD Date de création 25/07/2023, dernière modification le 25/07/2023