Integers occurring in the coefficients of the Taylor series of 1/cosh t
Not to be confused with Eulerian number or Euler's number.
For other uses, see List of things named after Leonhard Euler § Numbers.
In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion
,
where is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:
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.
In mathematics, the Eulernumbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e...
topology) – see Seifert fiber space Lucky numbers of EulerEuler's constant gamma (γ), also known as the Euler–Mascheroni constant Eulerian integers, more...
the natural numbers. Euler found that this sum equals exactly π2/6. Euler has also been credited for discovering that the sum of the numbers of vertices...
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k...
in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given...
The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field....
The numbers An are variously known as Eulernumbers, zigzag numbers, up/down numbers, or by some combinations of these names. The name Eulernumbers in...
combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These...
algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant...
Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether...
full-scale Reynolds number, and similarly for the Eulernumbers. The model numbers and design numbers should be in the same proportion, hence p m ρ m v...
prime numbers, then 2n × p × q and 2n × r are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case m = n − 1. Euler's rule creates...
primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from...
even perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if...
as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler (Letters to a German Princess) in 1768. The idea was popularised by Venn...
Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\...
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product...
mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.[under discussion] Some define the natural numbers as the non-negative...
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