The function euler computes the nth Euler number, or the nth Euler polynomial in x.

The nth Euler polynomial E(n, x) is defined by the exponential generating function:

The nth Euler number E(n) is defined by the exponential generating function:

The nth Euler polynomial E(n, x) and the nth Euler number E(n) are related by the following equation:

$\mathrm{euler}\left(6\right)$

$\mathrm{-61}$

$\mathrm{euler}\left(2\,5\right)$

$20$

$\mathrm{euler}\left(4\,x\right)$

$x^4-2x^3+x$

$\mathrm{euler}\left(4\,\frac{1}{2}\right)$

$\frac{5}{16}$

$\mathrm{euler}\left(2\,5x\right)$

$25x^2-5x$