# Euler Numbers, E(n)

Euler numbers, E(n), are integers and are defined for all n >= 0. E(2n+1) = 0, E(4n) > 0 and E(4n+2) < 0 for n = 0, 1, 2, ... .

E(0) = 1
E(2) = −1
E(4) = 5
E(6) = −61
E(8) = 1385
E(10) = −50521
E(12) = 27,02765
E(14) = −1993,60981
E(16) = 1,93915,12145 (11 Digits)
E(18) = −240,48796,75441 (13 Digits)
E(20) = 37037,11882,37525 (15 Digits)
E(22) = −69,34887,43931,37901 (17 Digits)
E(24) = 15514,53416,35570,86905 (20 Digits)
E(26) = −40,87072,50929,31238,92361 (22 Digits)
E(28) = 12522,59641,40362,98654,68285 (25 Digits)
E(30) = −44,15438,93249,02310,45536,82821 (27 Digits)

To compute E(n), n >= 0, if n is odd, E(n) = 0, if n is even:
E(0) = 1,
for n = 2, 4, 6, ...
Set m = n/2 − 1. Then
E(n) = −n! * Sum{j=0,m}[E(2j)/((n−2j)!(2j)!)]

For example
E(2) = −2! * (E(0)/(2!*0!)) = −1
E(4) = −4! * (E(0)/(4!*0!) + E(2)/(2!*2!)) = −1 + 6 = 5
E(6) = −6! * (E(0)/(6!*0!) + E(2)/(4!*2!) + E(4)/(2!*4!)) = −1 + 15 − 75 = −61
. . .

Here are some notes from my program XICalc - Extra Precision Integer Calculator http://www.oocities.org/hjsmithh/download.html#XICalc :

GenEuler(x) => Generate and save Euler number upto E(x):

The exact integer value of all even indexed Euler numbers up to E(x) are generated and saved in computer storage. If x is less than 4, it is taken to be 4. If x is an odd positive integer, the next even integer is used. If x is very large, an error message is generated: "GenEuler: n > 10000000, too large for Euler number". But it would take "forever" for an x = 10,000,000.

Euler(x) = E(x), Euler number x:

This function returns the exact integer value of the Euler number E(x). If x < 0, 0 is returned. If x is too large, 0 is returned and an error message is generated.

Here are some notes from my program XPCalc - Extra Precision Floating-Point Calculator http://www.oocities.org/hjsmithh/download.html#XPCalc :

Euler(x) = Euler number E(x):

This function returns the value of the Euler number E(x), an integer. If x is less than zero, zero is returned. If x is not an integer, the integer portion of x is used. E(0) = 1. E(1) = 0. E(2*n+1) = 0 for n >= 0. If x > 4.05997,83000,01705,888E+18, an alarm message is displayed. If x < 150 + 0.55 * decimal-digits, this is the same as the EulerN(x) command. If x is larger than this, the equation E(n) = (−1)^(n/2) * 2 * Gam(n+1) * Beta(n+1) / (Pi/2)^(n+1) is used.

EulerL(X) = Generalized Euler number E(X) for large |X|:

This is the same as the EulerG(X) function, but uses BetaL(x) to be faster for large |X|:

EulerL(x) = 2 * BetaL(−x).

EulerG(X) = Generalized Euler number E(X):

This is the analytic continuation of the Euler number. Good for all values of x (even complex x). If x is real and x >= 0 and x is an integer this is the same as Euler(x), otherwise

EulerG(x) = 2 * Beta(−x). See: Euler Number -- From MathWorld
And: Wolfram Function Definition -- EulerE

This page accessed times since April 3, 2005.
Page created by: hjsmithh@sbcglobal.net
Changes last made on Friday, 27-Mar-09 15:59:47 PDT