# Euler numbers

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##### 1: 24.1 Special Notation
###### EulerNumbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
##### 4: 24.14 Sums
###### §24.14 Sums
24.14.4 $\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(% 1-2^{n+2})\frac{B_{n+2}}{n+2}.$
##### 6: 24.5 Recurrence Relations
###### §24.5 Recurrence Relations
24.5.4 $\sum_{k=0}^{n}{2n\choose 2k}E_{2k}=0,$ $n=1,2,\dots$,
###### §24.5(iii) Inversion Formulas
$b_{n}=\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}E_{2k}a_{% n-2k}.$
##### 7: 24.20 Tables
###### §24.20 Tables
Wagstaff (1978) gives complete prime factorizations of $N_{n}$ and $E_{n}$ for $n=20(2)60$ and $n=8(2)42$, respectively. …
##### 8: 24.2 Definitions and Generating Functions
###### §24.2(ii) EulerNumbers and Polynomials
$E_{2n+1}=0$ ,
##### 9: 24.9 Inequalities
###### §24.9 Inequalities
24.9.7 $8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}% \right)>(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.$
24.9.10 $\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}.$
##### 10: 24.11 Asymptotic Approximations
###### §24.11 Asymptotic Approximations
24.11.3 $(-1)^{n}E_{2n}\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},$